Understanding Circular Measure in Mathematics

hey what's up you guys thank you for joining me again now in this video we are going to be covering the first chapter of form 5 admits which is circular measure so without further Ado let us get started [Music] so as usual before we start with the questions let us go through some of the examples or the notes that you have to know before you start with the examples okay so the there are four things that you have to know for this chapter the first thing is conversion you want to convert between Radian and degree what is radian and degree so I'm sure you all learned before angle right so angle you have different degrees so for example this is 30 degree let's say okay so this is 30 degree but there is another form of expressing degree which is called radian okay so radian is another form of expressing angle okay so is it similar as length length you have CM you have mm you have km right you have meter all of these are different units of length so same thing for angle you have degree and you also have radian so how do you convert between degree and radian so radian is usually expressed in the form of pi okay so how we usually do is we know that 360 degree is equivalent to 2 pi okay so this is in terms of degree is in terms of Radian so you usually write usually right there Pi 2 pi red red means radian okay short form for radian so what about 180 degree so 180 degree means half of this right so it's half means it's going to be pi so Pi can be some people might use 22 over 7 some people might use 3.142 some of y'all might even use the calculator here the recurring number you can also use either one okay unless the question state which one you're supposed to use okay other than that you can just it's up to you it's your choice and then what about 90 degree if it's 90 degree it's pi over 2 right it's another half of half of this so pi over 2 and if let's say I want 45 degree it becomes pi over 4. so basically these are the basics that you have to know okay but what if you get some funny angle like for example 60 degree or 60.5 degree how are you going to convert so how you convert is if let's say you want to convert from Radian to degree or maybe I do degree to Radiance okay it's the most common one degree to radian you have to this one is in terms of degree right so let's say 60 degree for example so degree you want to convert to radian you take 60 times pi over 180. pi over 180 this is to convert from degree to radian if you want to convert from radian to degree you just flip the uh fraction here okay this fraction here just flip this so you times 180 over pi okay times 180 over Pi so this is basically how you convert between radian and degree okay one is times 180 over Pi but the other one is times pi over 180. so you just have to decide which one you want to convert from the second thing you have to know is finding for arc length so this usually you express this as SL okay so what is upline let's say you have a circle okay you have a circle and you want to cut you want to calculate the arc length of a circle so let's say a full circle right this is what we call as circumference okay circumference but what if you just want to find one of the sector only let's say I don't want to find the circumference I just want to find here this sector here okay you want to find the length that means this arc length Okay Arc means basically the the circle one small part of the circumference is called Arc okay so you want to find this part so this not this not area by the way it's not area maybe I colored wrongly this is not area we are looking for the up length so after length you the equation is s equals to R Theta what is Theta Theta is the angle here R is the radius okay and this Theta here you have to use in the form of Radian okay you must use the angle you use must be in radian form you cannot use degree okay if you use degree you get a different answer so this is the formula for Arc Length s equals to R Theta what about the third point the third point you have to know is area of sector area of sector so area of sector see the previous one we learned about Arc link now what if you want to find the entire area if you want to find the entire area the formula is 1 over 2 R square Theta okay same thing the Theta here should be Radian the unit must be radian okay so what is Theta Theta is the angle here and then the r is the radius okay the last one you have to know is number four this thing is called area of segment oh by the way guys this formula is given to you this formula is also given to you the this formula here which I'm about to show you area of segment this one you have to memorize because it's not given to you okay so area of segment what's the formula by the way what is segment segment is like this imagine you got a sector here this vector the segment is when you draw a line between this point and this point okay key thing it has to be from this point to this point draw a line here this thing this place here is called the segment okay this thing is called the segment please take note the segment is between this corner to this corner if let's say the Act the question draw like this no and then they ask the sheet here this is not the segment because it has to be from one of the corner to the other Corner okay so this is not a segment if you draw like this okay then this is a segment okay so how do you find the area of this segment there are two ways you can do the first is you can use your manual way which is you find the whole thing the total area minus the triangle minus the triangle that's one way you can do the second way you can do is you can sorry my mouse are really not moving okay yeah the second way you can do is using this formula that I'm going to show you is by taking one over two R square Theta minus sine Theta okay so what is Theta like I said is the angle and then R is of course the radius now key thing you have to understand is this Pi here is called is using inter in the form of Radian but this Theta here is used in the form of degree okay make sure you take note of this okay whenever there is sine cos tangent anything to do it trigger you cannot use radian why because you get a different answer okay so if you don't believe right you can try this just try this take one of these okay or you take 45 degree for example take 45 degree and you can try if you type sine 45 and you type sine pi over 4 do you get the same answer and says nor you won't get the same answer and the correct answer is going to be this okay because whenever you are finding for using sine cost tangent Trigo you have to use in degree form okay so if 45 degree please use 45 degree don't use pi over 4 you'll get the wrong answer okay so sign remember sine cos tangent please use uh degree form but if it's not sine cost engine you can use radian okay so in this case remember this one is in gradient this one is in degree Okay so yeah that's about it these are the four things that you have to know uh to understand this chapter and yeah that's all so let us go straight to the examples all right so before we start off with the difficult questions let us start with the easy one first okay so in this case convert each of the following angles into degrees certain now you see they gave you the PI right so you cannot use 22 over 7 and you cannot use the calculator and you have to use 3.142 okay if you stated to you please use what they are what they have given so in this case you want to convert from Radian to degree so what do you use your times 180 over pi okay so in this case first one question a pi over 8 times 180 over pi so actually you can just cut this in the end you should get 22.5 degree okay next question B 1.04 radian you want to convert U times 180 divided by pi so your answer should be 59.58 degree okay you can leave your answer like this in the decimal form okay but some people they like to use uh York and convert to degree minute and second okay if you don't know that don't worry okay just use as it is but if you know you want to explore that in your calculator you see one of the button is like this one okay but of course depending what calculator you use by most likely going to be this button okay it's uh deep below the square root below the square root button okay if it's if you're using a different one and it's somewhere else you just look for this okay and this is the button so this one is your converting the degree into degree minute and seconds okay so if you convert that your answer should be 59 degree 35 seconds uh 35 minutes okay so this is your answer but if you if you're not using this even this is good enough okay right next in each of the following diagram poq is a sector of a circle with Center o convert each of the angle poq into radian so this one is converting from um angle degree to Radian so how do you convert you multiply pi over 180 why because you're converting the radian I told you before radian means in the form of Pi right so let's say you always multiply by the pi first okay in case you cannot remember that's how I remember okay so now first question I mean a um 150.5 degree if you want to convert to radian you must multiply by pi over 180. answer is key in your calculator okay you get the answer question B 220 degree multiply by pi over 180. you will get 3.8402 Radian okay so these are your answers okay all right next okay now you're getting a little bit more challenging okay but no worries you can do it don't worry so the diagram on the right shows a sector width okay so they gave you a sector with Center o and radius 7 okay given the arc length is 14 cm find the first thing they want to find is the angle and they want in degree they want in degree so take note of that question a they give you the Arc Length right so what's the formula for Arc Length s equals to R Theta okay so what is s s is 14 radius is 7 Theta you're looking for Theta okay you don't know what's that so Theta the angle is going to be 14 over 7 which is equals to 2. Radian okay so 2 Radian you have to convert to what degree right because they ask for degree so please take note of that so to degree how to convert to radian you take two times uh you want to convert to degree right so 180 divided by so your answer should be one one four point five eight okay in degree form or you can use the in terms of degree meaning and second is going to be one one four degree 35 seconds minutes okay this book this answer is given to you in your in your book okay if you use your textbook this is the answer okay but other than that you can just use in decimal form it's fine all right question B the perimeter of the Shaded segment okay they want to find the perimeter of this okay another area perimeter so you already found you already know the this part now all you left is here so how do you find that so basically there are many ways you can do but I find the easiest way is of course you draw your triangle first okay here is 7 cm what you can do is you can draw a triangle right here's the 90 degree okay so you know that the angle this angle is going to be half of the Theta that you found here this angle here is going to be half of the Theta that you form and then you can use three go to find this length and then you times 2 you'll get the full length so never let us do this together okay so what you can do is decides what opposite this this will be opposite right opposite end hypotenuse so you can use sine so sine so the angle here is what the angle here you are looking for is half of this angle right half of that so it's going to be 114.58 divided by 2 okay sine the angle equals to opposite which you're looking for over hypotenuse which is seven okay so you shift you get your opposite angle is going to be equal to 5.8898 this is in cm okay so if you want to find the perimeter how do you find the parameter parameter is equals to 14 cm Plus 2 times 5.8898 okay so if I draw it like this you're taking this part plus this part and then plus 14 okay so your final answer should be 25.7796 the diagram shows the chain linking the front and back cranks of a bicycle it is given that the circumference of the front and back cranks are 50 okay so which is the front and which is the back so 50.8 should be the bigger one so here's the front okay and then the back should be 30 point uh 30.5 cm and the front is 50.8 okay calculate the length of the bicycle chain so since they gave you the circumference you can find the you can find out the radius okay because if you want to find parameter here you already have the parameter these two now you need to find this part this one and this one okay so you know you can use S equals to R Theta okay Theta is given to you that 160 and 185 okay but of course you have to change the radian but now what is left is your radian I mean your radius sorry so you want to find the radius so the only way to find the radius is by using the circumference that they gave you okay so how do you solve that so you know that circumference is um the total angle for circumference is what 360 is what 2 pi right 2 pi radian correct so we're going to use this okay so s equals to so let's take the front versa is the front maybe I use the background okay following order so back the back crank and also the front okay so the back you know that s equals to R Theta so what is s the circumference is 30.5 equals to radius you are looking for radius I put your RB at the back times the 360 360 is about 2 pi right so you're gonna find RB is equivalent to 30.5 divided by 2 pi so you should get 15.25 over Pi okay I'm just going to leave it in terms of Pi so that I don't disturb my final answer okay I'll leave it as it is first next you want to find the front so front is going to be the same thing 50 0.8 equivalent to RF because the front multiplied by 2 pi so RF is equals to 50.8 divided by 2 pi so your answer is 25.4 over Pi okay you will see why I leave it in terms of Pi okay you see y later so now you already found your radius now you want to find the this part The Arc Length at this particular Arc okay you want to find this link so to find the first one let's say if we do the back first okay it's the back 160 you must convert it to radian 160 convert to radian is times pi over 180 okay and then the 185 you also have to convert okay you know what I'm not going to convert it there I'm just going to do all one shot okay so you want to find the the Arc line right so s of the back one okay equals to radius is 15.25 over pi times the angle so the angle is what 160 times pi over 180. so what you notice that you see here here go Pi here go Pi I can cut right it makes it very easy so I try not to calculate Everything at Once okay I try to make it in the form of Pi verse and then you see where you can cut okay so in this case when you use your calculator you should get your answer 13.5556 same it is the The Arc for um the back wheel a back crank and then you have the front so the arc for front is going to be um R is 25.4 over pi multiplied by so it's 185 times pi over 180. so same thing you see the pi here you can cut so in the end you should get your answer 26.1056 cm okay so they're asking for what calculate the length of the bicycle wheel right so it's basically asking for perimeter okay Lang of bicycle chain okay is the parameter equals to what 25 plus 25 plus 13.556 Plus 26.1056 so you get your final answer 89.6612 cm okay so this is your answer right next area of sector of a circle okay now same thing they gave you the the this circle here with the sector the radius is 3cm and then they say the minor Arc Length so this is the minor arc length is 5 cm now they ask you first question they want you to find m-o-n so m o n is here okay they want you to find the angle but in the form of degree okay so same thing this is what we did before previous question they give you the Arc Length so you just use the formula s equals to R Theta so s is what the arc length is 5 equals to R is three Theta so what is Theta theta equals to 5 over 3. here you can leave it like this or you can write in um decimal form so decimal form would be I mean 1.6667 but I try not to leave it in a decimal form first because because whenever you round round off the number right your final answer will change so I try not to do that okay so but now they were in degree right so you have to change it to degree first so you times 180 over pi so you will get your answer is 95.48 degree Okay so you're going to need both of this later okay why because you see the second question what they're asking area of shaded segment so this this part right is the segment so shaded segment is I told you all the formula before question B area of segment is what area of segment is equals to 1 over 2 R square Theta minus sine Theta and this one is in radian don't forget this one is in degree okay please don't mix up so 1 over 2 what is your R radius is 3 right so 3 Square and then let's see here Pi is your sorry your Theta is in radian so radiant nearly found here right in terms of radian 5 over 3. minus sign what's the angle there in degree right so this is your in degree form so 95 .48 okay so you just key this into your calculator you'll get your answer straight away which is the sorry 3.0206 cm square okay here yeah that's about it so it's quite straightforward right once you know the formula is quite easy okay okay let's go to more challenging question so now this one the diagram on the right shows a cross section of a water pipe with internal radius okay so they specifically said internal radius so that means the radius is from the center all the way to here this part okay it's not touching the it's not the outside part here it's not including that it's just the internal area so it's 12 CM and then they ask you to find the height so the the H here okay and the horizontal width is horizontal width is here okay 18. so the first question they ask you to find H okay so how do you find the h okay the first thing you want to understand is if you have a sector like this okay how am I supposed to find this you take the radius because you know that since this is the radius right so that means from here to the whole thing down here the whole length is radius correct so if I want to find the height I must take the whole radius minus this part let's say this x okay this part from here to here only you can see you know right I think the whole length which is the radius minus this part I'll get the height correct so that's what we're going to do we're going to sorry am I drawing is pretty bad but that's basically that's what you're going to do we are going to find the we already know the radius now all we have to know is the height of that triangle so let me redraw this actually you can see clearly so you want to find this height so let's assume this height is let's produce X okay so the whole the whole link between EF is 18 right so if I cut into half it's going to be 9 here you're going to be 9. correct nine plus nine eighteen and then radius is 12. so how do we find X okay I I don't access the height of the triangle okay so X is pythagore's theorem okay so x equals to square root 12 squared minus 9 squared you should get three set seven okay so I'm going to leave it in this like I said I want to round round off the number so quickly so I'm going to use three seven okay CM now you want to find the height right height is here so the height is equivalent to the radius which is 12 minus 3 7. so your answer is going to be 4.0627 cm okay this is the height next question they ask you to calculate cross-section area covered by water so basically they're asking for area of the segment correct you see where is the water in the diagram this part right so it's the area of the segment so how do you find the area of segment the formula correct format sorry question B formula is what area of segment one over two R square Theta minus sine Theta okay the first Theta is in terms of radian the second Theta is in terms of degree okay so now how are we going to find the angle huh so we know that we are looking for this angle right this whole angle is what the whole angle is Theta but now what I can do is I can use this triangle to form find this right and I times 2 I will get two sides correct so I'm basically using the triangle so I can use one socatoa right so if this whole thing is pi sorry if this whole thing is Theta the half here is called Theta over 2 and here so Theta over 2 correct so let me redraw this so you got a triangle now case 12 he is 9 here is Theta over 2. okay because the whole thing is Theta so the half is only Theta O2 okay so now we can use socato so in this case what are we going to use opposite and hypotenuse so it's sine right so sine Theta over 2 equals to opposite is 9 over 12. okay so you can shift this you should get your Theta is equivalent to 97.18 degree so this is in degree but you also need in terms of Radiance times pi over 180. so you should get your answer 1.6963 radian okay so one in terms of degree one in terms of radian now you can find your area of segment one over two times R what's the radius radius is 12 12 square multiply by Theta this versatiles in terms of radian right so 1.6963 minus sign this is in terms of degree so 97.18 okay key this into your calculator you should get your answer straight away 50.7019 cm square okay so this is how you solve much challenging questions so I hope you guys understand this okay now let's see more examples so the diagram on the right shows two discs with radi 11 cm and 7 cm touching each other at point R okay that these are on a straight line p d c q calculate b a d in degree okay so let me redraw these are you got basically like a trapezium right so here 90 degree here 90 degree here is 7 cm here total is 11 cm okay so now they want you to find bad where's bad here is a CE B here is D here you see so bad should be the angle here okay so how in what way can you find that so I want your I want you all to understand this there's a trick usually right when there is a 90 degree there is always a right angle there is always a right angle okay so where is your right angle in this diagram the right angle is here you can draw a line parallel to the bottom line here okay so that means here is also 90 degree so now you have your right angle triangle okay you have your right angle triangle so if this is a right angle triangle we know this is 7 cm because it's going to be the same as this right so what is the length here 11 minus 7 so here is 4. okay you want to find this angle you know these four what other data can you get you know you know this length you should know this length how to know because you notice that in this diagram they are the radius correct this radius here is 11 and then this radius here is seven so that means this length a b is actually 11 plus 7 which is 18. okay so now you have this length you have this you have this so you can use your soccer okay to find the angle uh bad so question a you want to find angle BD so what what are you going to use you're going to use um a Json and hypotenuse so it's cos okay so cos angle b a d equals to adjacent is 4 hypotenuse is 18. okay so what's your angle bad should be equals to 77.16 degree okay so they want in degree right so this answer question B subsequently find the Shaded area so shaded area is where actually if you don't didn't notice this is the Shaded area okay this is the Shaded area so how are you going to find this area the only way you can find this area right is by using the whole trapezium here find the total area minus this area the area of this sector and minus the area of the sector okay that's it when you can do that you get your answer shaded area so area of trapezium you can find because you know the formula is what one over two the two parallel sides you add them up a plus b times the height okay so the height is here but you do know what is the height correct so how are you going to find the height the height is going to be you can use back this triangle right you know this and you know this so to find the length CD you just have to do Pythagoras Theorem okay to find okay let's do it together so question B okay I'm just gonna draw here so there you go space question B so let us first find the height here okay so let's assume this is the height so height is equals to 18 square minus 4 Square and then square root okay so your answer should be 2 3 77 okay I'm gonna leave it in this form I don't want to round off first if I write in decimal I have to round off so I don't want that to do that so early in the process so this is 277 cm so you can now find the area of the entire uh trapezium so now you have to find area of the SEC sector yeah so this sector you can find because you already know this first sector here this one you can find because you know the radius because the formula is what 1 over 2 R square Theta right so Theta is the angle radius R is radius so you already know the radius and you know the ta the angle which you already found here in the 77.16 but you have to change to radian later the second sector you have to find is this way right but this one is a bit hard because first thing is you don't know uh X actually you already know the radius which is 7 cm but you don't know the angle right you don't know what's the Theta so now you have to find the Theta first so how are you going to find the Theta how are you going to find so the only way you can find is actually maybe there are other ways left but the only the way that I can think of is we can take you know that a full trapezium What's the total angle 360 right here is 90 here is 90. here your default is 77.16 so now you can find this right correct so you can just straight away do that to find the angle so we are going to do that okay so let me just erase this okay all right so let's find uh angle um angle a b c okay so that angle is going to be equals to 360. minus 90. minus 90 minus 77.16 so that angle is equals to 1 0 2.84 degree okay so this angle of course uh we have to change to radian because we want to use to solve the area okay so now we're going to straight away do that area of shaded region okay area of Shield region is equals to you take the whole area right area of the trapezium so one over two times a plus b what is a and what is b a is seven the two parallel sides are please don't take this and these are Noah must be the parallel sides so the parallel sides are seven and the other side is 11 so 7 plus 11. multiply by height is what you recalculated 277 okay minus area of the first sector there which is 1 over 2 R square is R is 11 squared Theta is what Theta is bad right so it's 77 .16 don't forget to change to radian so pi over 180. and then minus the second sector 1 over 2 R square R is seven square and then the angle is um one zero two point eight four so one zero two point eight four times pi over 180. okay so use your calculator you should get 18 77 here minus 81.4857 minus 43.9807 so your final answer should be 32.4830 cm square okay so that's how you solve this kind of question all right next the diagram on the right shows a moon shaped kite whose line of symmetry is OS okay for this question right I really need you all to see closely because it's quite tricky okay so just look at this huh so aqb where is aqb this aqb right aqb is Arc sector Arc of a sector from from a circle with Center also sorry so here is a sector okay I need your take note because if you don't see closely which Arc it belongs to and you will get you will in the end right you don't know which radius to use okay so this question is a bit Messina so I need you all to take note of that so now in this case this Arc what is the radius radius is 20. the center is O Okay so take note of that the second one they say is what a p b r a base a p b r a p b r is a what semicircle so that means for this Arc this Arc here if you want to find this Arc you need the radius is 16 okay please don't use the wrong radius if you use 20 you get the wrong answer so okay so that's done next they tell you radius 16 Tru is an arc from Circle with Center as so this one is quite straightforward this arcade belongs to this sector okay and then the radius is 12 and then given that Arc Length of Tru so this outline here is 21 cm calculate aob and TSU where is aob aob is here TSU is here so TSU is actually quite easy right this is quite easy okay let's find TSU first huh so TSU you can just use the formula s equals to R Theta so the S is 21 R is 12 Theta is Theta so theta equals to 21 over 12. which is 3 oh sorry 704 so right here seven over four okay I try not to simplify this so that um in the end later um anyway this one I think you can do it with this one you can simplify to 1.758 so I just gonna write 1.75 1.75 radian if it's a recurring decimal I try not to simplify so soon okay because otherwise the final answer will be will be very far from the actual answer so now this is Tsui angle t s u now you want to find angle aob so angle aob let's say this Theta okay so this you know this is 20 this is 16 so you can form your right angle here right so this small angle here is Theta over 2 okay because it's you cut into half so Theta over 2 is that's half of that angle so we're going to use um opposite this angle here is of this length is opposite this hypotenuse right so opposite hypotenuse is sine so sine um Pi will give it sine Theta over 2 okay I cannot use Theta because you overlapped with the this previous question here this one so I'm not going to use Theta let's just use angle aob over 2 okay sine of this is equals to opposite which is 16 over adjacent so you shift one by one you should get angle A or B equals to 1 0 6.26 degree make sure you change to radian because it asks for Radian times pi over 180 you will get 1.8548 Radian okay so this is um the two angles that you're looking for okay next they ask for perimeter so where's the parameter perimeter here you already know 21 here you also know 12 and 12. so all you have to find is this and this okay when you can find the two then you can find the perimeter so let's find um question Bia let's first find ARB so ARB is equals to so we use this as a semicircular at least so I say you have to know which radius to use okay so since it's a semicircle is going to be um R Theta right R is 16. Theta is the angle is half a circle so I told you 480 is equals to PI right in radian form right so this is pi so you get 16 pi I'm just going to write 16 pieces okay CM after that you're gonna find for aqb aqb is using the sector okay so the sector the radius is 20. Theta is you found the aob right so it's this one so it's going to be 1.8548 so you get your answer is 37.0966 CM so you want to find perimeter right so I'm going to write here perimeter equals to 37.0966 plus 16 pi plus um 12 plus 12 plus 21. okay your final answer should be one three two point three six eight six cm so now we have to find for the last question which is area so how do you find area so area here is a bit complicated so yeah okay let me erase this because I don't have enough space okay here is this and this one okay now we want to find for area this area is very easy to find right because you just use the formula however if you want to find this other shape here this moon shape is a bit hard because why first thing is there is no formula for this second thing is you have basically two shapes involved first is a semicircle and a sector so take note now how are we going to do this we are first going to take the semicircle yeah use a different color we take the semicircle now the semicircle you have to minus this correct to find this area so how do you find the area of that segment you can use the formula area of segment formula but in this case I'm not going to do that what I'm going to do is I'm going to take the entire area of the sector entire area of the sector minus this triangle because we know it's a triangle right so I'm just going to minus that also again okay you can use that or you can use the formula of the segment area of segment formula okay so I'm going to use the way that I just mentioned minus a triangle so I mean let's do this together so Ariel question C here area is equals to let's find this first this one so it's very straightforward 1 over 2 r squared what is r 12 square um Theta is where is TSU okay here 1.7 5. plus okay now here is where it gets tricky so you want to First find the semicircle first right so semicircle is 1 over 2 r squared what is r r is 16. Square um Theta okay I don't know whether you can see or not because I color okay so are the radius is 16 and then Theta is one half a circle is pi okay minus so the semicircle must minus the sector correct but the sector is what how to find the sector the sector you have minus 1 over 2 um our R is 20. r squared theta wasted is um one point eight five four eight minus the triangle so triangle is one over two times base basis 16 plus 16 is 32 it is the beta okay or what about the height okay so the height here is op how to find op you can use this triangle okay so op is equals to 1. 20 squared minus 16 square a Pythagoras Theorem so you should get 12 okay so that is 12 CM so 1 over 2 times 32 times height is 12. then you close all the bracket okay I know it can be a bit confusing and I'm sorry I'm writing it this way because it's uh so hard to see but as long as you understand what I'm trying to do okay so in the end you should get so this part here you should get it is the area of t r u s correct this sector here so that area here should be one two six Plus so this area of the semicircle area of semicircle versus semicircle a p b a p b r yeah apbr should be four zero two point one seven six okay minus minus what should be the sector the sector should be what is the sector again A O B Q right aob Q is 370. 96 minus the triangle which is 192. okay so in the end you can use your calculator you should get your answer is three four nine point two one six cm square oh my God I hope I hope it's not confusing okay okay I'm gonna repeat this okay because I don't want you all to get confused let me erase all this okay so basically what is happening is you're gonna find this area which is this part okay area of the t r u s okay plus area of the semicircle semicircle is um AP B R okay minus you want to find the sector so how to sorry you want to find the segment so how to find the segment the segment is you have to first find the sector minus the triangle up here okay so the sector is um a o b q minus the triangle triangle aob AO B pill okay so basically this is what I'm doing here okay so the trus okay this sector is this one and then I Plus apbr which is the semicircle so this is apbr minus this one is to find the area of segment which is what I did here this part okay so I hope you find this uh more and able to understand okay so I hope I didn't confuse you guys all right let's see the next question we have two more questions and we're done with the chapter so let's see this in the diagram on the right are three identical 20 cent coins with the same red eye and touching each other if the blue colored region has an area of okay so we give you the area of the colored region find the radius of each coin okay so they give you nothing no data except the area of um the color region so how are you going to find so if you notice is a tricky question but what you can do is here there's a center here correct there's a center here for each of the coin what you can do is you can connect the radius here is the radius for this coin and here's the radius for this coin here's the radius for this coin [Music] and here's the radius for this coin the radius here and the radius here Okay so here is R here is R here's R here's our his R user because all same radius right same coin same radius so in other words you have a triangle okay you have a triangle here is 2R here is 2R here is 2R okay so since they are all same length or two are right so it's an equilateral triangle and for an equilater equilateral triangle what are the properties so the property is all three angles are the which is 60 degree okay 180 day battery so you get 60 degree each of the angle so now we can use this info to solve okay so what is the formula we want we can to find the radius right we have to form an equation using this okay because only this is given to you so how to find the area of colored region how do you find that area the area of the color region is equal to area of the triangle minus three times of the area of sector okay so in other words you just have to find each sector okay if we just find one sector times three you'll get three sectors so the whole area of triangle minus the three sectors you should get the area of the colored region so let's form the equation okay so color region is given to you 12.842 area of triangle so you can use 102 times base times height but in this case the easier one would be we learn from four area of triangle one over two a b sine C okay remember I think I told you all Chopstick right like Chopstick so a b sine C so the two length if let's say I'm using this angle right so the two length must be this two length okay if I'm using this angle the two length should be these two length okay so 1 over 2 a B is 2R times two r sine C so a 60 degree so make sure you 60 degree because this one is sine cos tangent remember I said sine cos tangent you have to use in degree okay if it's uh the normal formula for this chapter then you use in radian so in this case this is the formula minus 3 times area of the sector so 1 over 2 1 over 2 times R is what the radius is um each of these R right so it's r Square times um the angle is 60 degrees so 60 please change it to radian so pi over 180. okay so you will get 1 2 8.3 12.842 equals to third tree R square minus pi over 2 R square and then you rearrange this you should get R square equals to um 79.738 okay so in the end your R should be equals to 8.9297 [Music] this is what mm mm yeah okay so the actual answer that was given is actually eight point nine three one so it's actually pretty close so it's acceptable okay so it doesn't this kind of this chapter right it you won't get a accurate like exact answer because there are many decimal places involved okay some people might just take three decimal place and then the pi some people might use uh 3.142 some might use 22 over 7. in this case I'm just using 3.142 okay uh some people might use the one in the calculator okay so you will in the end will get different answers okay yeah so this is how you solve this question don't be uh frightened by the question okay try to figure out how can you solve this because only one value is given so you must try to figure out how you can solve this okay all right one last question let's go [Music] the diagram on the right shows the logo of an ice cream company the logo is made up of three identical sector okay so these are the sectors here the three sectors okay from the center and radius of 30 cm so all these are the radius 30 cm30 same because all identical right and then it's given that all these angles are 60 so here 60. here's 60 calculate equation a they ask you to calculate up length of a b raise a b a b is here so what's the formula s equals to R Theta so I'm just going to write a b a b equals to radius is 30. Theta is 60. please change it to Radian okay so you should get your answer 3 1.42 cm okay that's the first one second okay I need to cram a bit because I don't think we got space here but I mean um second one is area of cod right cod cod is here they're asking for sector right so you have to include the segment as well so this whole thing okay so we just use the formula of sector so area of sector Cod is equals to 1 over 2 R square what is r 30 square and then Theta is 60 times pi over 180. so your answer should be 150 Pi also equals to 471 .3 cm square okay so if you get anywhere close to 471.15 is acceptable okay that's the actual answer next you want to find for perimeter of segment EF where is EF okay EF is here so this is a perimeter so the outline you already found right outline is this because this identical shape so here is 31 point 31.42 yes so now you want to find the EF this straight line here how to find that so what you can do is you can form your try you can see your triangle right your triangle oops sorry okay so this is e f so you can form your triangle here 90 degree okay so what you can do is here's 60 degree right so this one is going to be 30. 30 degree you know the radius is 30 cm you want to find here okay so what is that angle that let's say this is um opposite right opposite so opposite and hypotenuse so you can use sokatoa so you can use a o o and H so sine so question three sine 30 is equivalent to opposite over 30. so what's your opposite angle opposite angle should be equivalent to 15 CM okay so here is 15. so now you need to find 15 and 15. here let me just right here so this half here is 15. here is 15 years of 15. right and then you want to find the parameter okay so perimeter is equivalent to 31.42 plus 15 plus 15. so your final answer is 61.42 cm okay next area of segment EF so this one I already gave you all the formula right just straight away use the formula no need to thinkery so area is equivalent to 1 over 2 r squared what is r the T Square Times Theta is 1 Theta is 60 so 6 you have to change to a radian 60 times pi over 180 minus sine 60 okay because this must be in 3 and this one must be in degree so you should get straight away you can get your answer 81.5886 okay now like I said it doesn't have to be 100 accurate okay as long as it's close to the answer because you might use different value of pi okay or you might use a three decimal place or two decimal place okay so in the end you'll get a you may not get the exact same answer but if it's close it's acceptable okay last two question um the logo is casted in cement if the thickness is uniform and it's 5 cm so the thickness now is 5 cm okay find the amount of cement needed in to make the logo so to do that you must first find the total area and then times the thickness okay the total area that means in 2D form of the logo times the thickness so what is the total area actually yesterday we find the total volume so total they ask for volume right equals to what's the area area of the whole sector right so you got Cod is this area one area right of the sector so you take three times of that which is three times four seven one point three and then you times 5 cm because that's the thickness so in the end you should get seven zero six nine point five is a volume okay so that's how you find the volume last question question C what are they asking if the cost of cement is 50 cents per CM Cube find the total costs okay so they are saying one cm Cube is 50 Cents right so how many you have how much you have you have seven thousand and 69.5 cmq is equivalent to this multiplier okay actually I shouldn't write equal I should just okay this one is 7069.5 times 0.50 so your answer should be RM three five three four point seven five okay so this is how you solve this kind of question it's actually pretty straightforward write this question it looks long but it's actually very straightforward all right guys so that's all for this video These are the four main things that we have covered in this video we have covered the whole entire chapter we have covered radian Arc Length Arc of the sector and we also covered out of segment and then some application questions all right so that is all for this video I hope you find this useful if you did don't forget to like subscribe and share it with someone who will also find this beneficial right so until next time hope to see you on my next video alright take care bye

hey what&#39;s up you guys thank you for joining me again now in this video we are going to be covering the first chapter of form 5 admits which is circular measure so without further Ado let us get started [Music] so as usual before we start with the questions let us go through some of the examples or the notes that you have to know before you start with the examples okay so the there are four things that you have to know for this chapter the first thing is conversion you want to convert between Radian and degree what is radian and degree so I&#39;m sure you all learned before angle right so angle you have different degrees so for example this is 30 degree let&#39;s say okay so this is 30 degree but there is another form of expressing degree which is called radian okay so radian is another form of expressing angle okay so is it similar as length length you have CM you have mm you have km right you have meter all of these are different units of length so same thing for angle you have degree and you also have radian so how do you convert between degree and radian so radian is usually expressed in the form of pi okay so how we usually do is we know that 360 degree is equivalent to 2 pi okay so this is in terms of degree is in terms of Radian so you usually write usually right there Pi 2 pi red red means radian okay short form for radian so what about 180 degree so 180 degree means half of this right so it&#39;s half means it&#39;s going to be pi so Pi can be some people might use 22 over 7 some people might use 3.142 some of y&#39;all might even use the calculator here the recurring number you can also use either one okay unless the question state which one you&#39;re supposed to use okay other than that you can just it&#39;s up to you it&#39;s your choice and then what about 90 degree if it&#39;s 90 degree it&#39;s pi over 2 right it&#39;s another half of half of this so pi over 2 and if let&#39;s say I want 45 degree it becomes pi over 4. so basically these are the basics that you have to know okay but what if you get some funny angle like for example 60 degree or 60.5 degree how are you going to convert so how you convert is if let&#39;s say you want to convert from Radian to degree or maybe I do degree to Radiance okay it&#39;s the most common one degree to radian you have to this one is in terms of degree right so let&#39;s say 60 degree for example so degree you want to convert to radian you take 60 times pi over 180. pi over 180 this is to convert from degree to radian if you want to convert from radian to degree you just flip the uh fraction here okay this fraction here just flip this so you times 180 over pi okay times 180 over Pi so this is basically how you convert between radian and degree okay one is times 180 over Pi but the other one is times pi over 180. so you just have to decide which one you want to convert from the second thing you have to know is finding for arc length so this usually you express this as SL okay so what is upline let&#39;s say you have a circle okay you have a circle and you want to cut you want to calculate the arc length of a circle so let&#39;s say a full circle right this is what we call as circumference okay circumference but what if you just want to find one of the sector only let&#39;s say I don&#39;t want to find the circumference I just want to find here this sector here okay you want to find the length that means this arc length Okay Arc means basically the the circle one small part of the circumference is called Arc okay so you want to find this part so this not this not area by the way it&#39;s not area maybe I colored wrongly this is not area we are looking for the up length so after length you the equation is s equals to R Theta what is Theta Theta is the angle here R is the radius okay and this Theta here you have to use in the form of Radian okay you must use the angle you use must be in radian form you cannot use degree okay if you use degree you get a different answer so this is the formula for Arc Length s equals to R Theta what about the third point the third point you have to know is area of sector area of sector so area of sector see the previous one we learned about Arc link now what if you want to find the entire area if you want to find the entire area the formula is 1 over 2 R square Theta okay same thing the Theta here should be Radian the unit must be radian okay so what is Theta Theta is the angle here and then the r is the radius okay the last one you have to know is number four this thing is called area of segment oh by the way guys this formula is given to you this formula is also given to you the this formula here which I&#39;m about to show you area of segment this one you have to memorize because it&#39;s not given to you okay so area of segment what&#39;s the formula by the way what is segment segment is like this imagine you got a sector here this vector the segment is when you draw a line between this point and this point okay key thing it has to be from this point to this point draw a line here this thing this place here is called the segment okay this thing is called the segment please take note the segment is between this corner to this corner if let&#39;s say the Act the question draw like this no and then they ask the sheet here this is not the segment because it has to be from one of the corner to the other Corner okay so this is not a segment if you draw like this okay then this is a segment okay so how do you find the area of this segment there are two ways you can do the first is you can use your manual way which is you find the whole thing the total area minus the triangle minus the triangle that&#39;s one way you can do the second way you can do is you can sorry my mouse are really not moving okay yeah the second way you can do is using this formula that I&#39;m going to show you is by taking one over two R square Theta minus sine Theta okay so what is Theta like I said is the angle and then R is of course the radius now key thing you have to understand is this Pi here is called is using inter in the form of Radian but this Theta here is used in the form of degree okay make sure you take note of this okay whenever there is sine cos tangent anything to do it trigger you cannot use radian why because you get a different answer okay so if you don&#39;t believe right you can try this just try this take one of these okay or you take 45 degree for example take 45 degree and you can try if you type sine 45 and you type sine pi over 4 do you get the same answer and says nor you won&#39;t get the same answer and the correct answer is going to be this okay because whenever you are finding for using sine cost tangent Trigo you have to use in degree form okay so if 45 degree please use 45 degree don&#39;t use pi over 4 you&#39;ll get the wrong answer okay so sign remember sine cos tangent please use uh degree form but if it&#39;s not sine cost engine you can use radian okay so in this case remember this one is in gradient this one is in degree Okay so yeah that&#39;s about it these are the four things that you have to know uh to understand this chapter and yeah that&#39;s all so let us go straight to the examples all right so before we start off with the difficult questions let us start with the easy one first okay so in this case convert each of the following angles into degrees certain now you see they gave you the PI right so you cannot use 22 over 7 and you cannot use the calculator and you have to use 3.142 okay if you stated to you please use what they are what they have given so in this case you want to convert from Radian to degree so what do you use your times 180 over pi okay so in this case first one question a pi over 8 times 180 over pi so actually you can just cut this in the end you should get 22.5 degree okay next question B 1.04 radian you want to convert U times 180 divided by pi so your answer should be 59.58 degree okay you can leave your answer like this in the decimal form okay but some people they like to use uh York and convert to degree minute and second okay if you don&#39;t know that don&#39;t worry okay just use as it is but if you know you want to explore that in your calculator you see one of the button is like this one okay but of course depending what calculator you use by most likely going to be this button okay it&#39;s uh deep below the square root below the square root button okay if it&#39;s if you&#39;re using a different one and it&#39;s somewhere else you just look for this okay and this is the button so this one is your converting the degree into degree minute and seconds okay so if you convert that your answer should be 59 degree 35 seconds uh 35 minutes okay so this is your answer but if you if you&#39;re not using this even this is good enough okay right next in each of the following diagram poq is a sector of a circle with Center o convert each of the angle poq into radian so this one is converting from um angle degree to Radian so how do you convert you multiply pi over 180 why because you&#39;re converting the radian I told you before radian means in the form of Pi right so let&#39;s say you always multiply by the pi first okay in case you cannot remember that&#39;s how I remember okay so now first question I mean a um 150.5 degree if you want to convert to radian you must multiply by pi over 180. answer is key in your calculator okay you get the answer question B 220 degree multiply by pi over 180. you will get 3.8402 Radian okay so these are your answers okay all right next okay now you&#39;re getting a little bit more challenging okay but no worries you can do it don&#39;t worry so the diagram on the right shows a sector width okay so they gave you a sector with Center o and radius 7 okay given the arc length is 14 cm find the first thing they want to find is the angle and they want in degree they want in degree so take note of that question a they give you the Arc Length right so what&#39;s the formula for Arc Length s equals to R Theta okay so what is s s is 14 radius is 7 Theta you&#39;re looking for Theta okay you don&#39;t know what&#39;s that so Theta the angle is going to be 14 over 7 which is equals to 2. Radian okay so 2 Radian you have to convert to what degree right because they ask for degree so please take note of that so to degree how to convert to radian you take two times uh you want to convert to degree right so 180 divided by so your answer should be one one four point five eight okay in degree form or you can use the in terms of degree meaning and second is going to be one one four degree 35 seconds minutes okay this book this answer is given to you in your in your book okay if you use your textbook this is the answer okay but other than that you can just use in decimal form it&#39;s fine all right question B the perimeter of the Shaded segment okay they want to find the perimeter of this okay another area perimeter so you already found you already know the this part now all you left is here so how do you find that so basically there are many ways you can do but I find the easiest way is of course you draw your triangle first okay here is 7 cm what you can do is you can draw a triangle right here&#39;s the 90 degree okay so you know that the angle this angle is going to be half of the Theta that you found here this angle here is going to be half of the Theta that you form and then you can use three go to find this length and then you times 2 you&#39;ll get the full length so never let us do this together okay so what you can do is decides what opposite this this will be opposite right opposite end hypotenuse so you can use sine so sine so the angle here is what the angle here you are looking for is half of this angle right half of that so it&#39;s going to be 114.58 divided by 2 okay sine the angle equals to opposite which you&#39;re looking for over hypotenuse which is seven okay so you shift you get your opposite angle is going to be equal to 5.8898 this is in cm okay so if you want to find the perimeter how do you find the parameter parameter is equals to 14 cm Plus 2 times 5.8898 okay so if I draw it like this you&#39;re taking this part plus this part and then plus 14 okay so your final answer should be 25.7796 the diagram shows the chain linking the front and back cranks of a bicycle it is given that the circumference of the front and back cranks are 50 okay so which is the front and which is the back so 50.8 should be the bigger one so here&#39;s the front okay and then the back should be 30 point uh 30.5 cm and the front is 50.8 okay calculate the length of the bicycle chain so since they gave you the circumference you can find the you can find out the radius okay because if you want to find parameter here you already have the parameter these two now you need to find this part this one and this one okay so you know you can use S equals to R Theta okay Theta is given to you that 160 and 185 okay but of course you have to change the radian but now what is left is your radian I mean your radius sorry so you want to find the radius so the only way to find the radius is by using the circumference that they gave you okay so how do you solve that so you know that circumference is um the total angle for circumference is what 360 is what 2 pi right 2 pi radian correct so we&#39;re going to use this okay so s equals to so let&#39;s take the front versa is the front maybe I use the background okay following order so back the back crank and also the front okay so the back you know that s equals to R Theta so what is s the circumference is 30.5 equals to radius you are looking for radius I put your RB at the back times the 360 360 is about 2 pi right so you&#39;re gonna find RB is equivalent to 30.5 divided by 2 pi so you should get 15.25 over Pi okay I&#39;m just going to leave it in terms of Pi so that I don&#39;t disturb my final answer okay I&#39;ll leave it as it is first next you want to find the front so front is going to be the same thing 50 0.8 equivalent to RF because the front multiplied by 2 pi so RF is equals to 50.8 divided by 2 pi so your answer is 25.4 over Pi okay you will see why I leave it in terms of Pi okay you see y later so now you already found your radius now you want to find the this part The Arc Length at this particular Arc okay you want to find this link so to find the first one let&#39;s say if we do the back first okay it&#39;s the back 160 you must convert it to radian 160 convert to radian is times pi over 180 okay and then the 185 you also have to convert okay you know what I&#39;m not going to convert it there I&#39;m just going to do all one shot okay so you want to find the the Arc line right so s of the back one okay equals to radius is 15.25 over pi times the angle so the angle is what 160 times pi over 180. so what you notice that you see here here go Pi here go Pi I can cut right it makes it very easy so I try not to calculate Everything at Once okay I try to make it in the form of Pi verse and then you see where you can cut okay so in this case when you use your calculator you should get your answer 13.5556 same it is the The Arc for um the back wheel a back crank and then you have the front so the arc for front is going to be um R is 25.4 over pi multiplied by so it&#39;s 185 times pi over 180. so same thing you see the pi here you can cut so in the end you should get your answer 26.1056 cm okay so they&#39;re asking for what calculate the length of the bicycle wheel right so it&#39;s basically asking for perimeter okay Lang of bicycle chain okay is the parameter equals to what 25 plus 25 plus 13.556 Plus 26.1056 so you get your final answer 89.6612 cm okay so this is your answer right next area of sector of a circle okay now same thing they gave you the the this circle here with the sector the radius is 3cm and then they say the minor Arc Length so this is the minor arc length is 5 cm now they ask you first question they want you to find m-o-n so m o n is here okay they want you to find the angle but in the form of degree okay so same thing this is what we did before previous question they give you the Arc Length so you just use the formula s equals to R Theta so s is what the arc length is 5 equals to R is three Theta so what is Theta theta equals to 5 over 3. here you can leave it like this or you can write in um decimal form so decimal form would be I mean 1.6667 but I try not to leave it in a decimal form first because because whenever you round round off the number right your final answer will change so I try not to do that okay so but now they were in degree right so you have to change it to degree first so you times 180 over pi so you will get your answer is 95.48 degree Okay so you&#39;re going to need both of this later okay why because you see the second question what they&#39;re asking area of shaded segment so this this part right is the segment so shaded segment is I told you all the formula before question B area of segment is what area of segment is equals to 1 over 2 R square Theta minus sine Theta and this one is in radian don&#39;t forget this one is in degree okay please don&#39;t mix up so 1 over 2 what is your R radius is 3 right so 3 Square and then let&#39;s see here Pi is your sorry your Theta is in radian so radiant nearly found here right in terms of radian 5 over 3. minus sign what&#39;s the angle there in degree right so this is your in degree form so 95 .48 okay so you just key this into your calculator you&#39;ll get your answer straight away which is the sorry 3.0206 cm square okay here yeah that&#39;s about it so it&#39;s quite straightforward right once you know the formula is quite easy okay okay let&#39;s go to more challenging question so now this one the diagram on the right shows a cross section of a water pipe with internal radius okay so they specifically said internal radius so that means the radius is from the center all the way to here this part okay it&#39;s not touching the it&#39;s not the outside part here it&#39;s not including that it&#39;s just the internal area so it&#39;s 12 CM and then they ask you to find the height so the the H here okay and the horizontal width is horizontal width is here okay 18. so the first question they ask you to find H okay so how do you find the h okay the first thing you want to understand is if you have a sector like this okay how am I supposed to find this you take the radius because you know that since this is the radius right so that means from here to the whole thing down here the whole length is radius correct so if I want to find the height I must take the whole radius minus this part let&#39;s say this x okay this part from here to here only you can see you know right I think the whole length which is the radius minus this part I&#39;ll get the height correct so that&#39;s what we&#39;re going to do we&#39;re going to sorry am I drawing is pretty bad but that&#39;s basically that&#39;s what you&#39;re going to do we are going to find the we already know the radius now all we have to know is the height of that triangle so let me redraw this actually you can see clearly so you want to find this height so let&#39;s assume this height is let&#39;s produce X okay so the whole the whole link between EF is 18 right so if I cut into half it&#39;s going to be 9 here you&#39;re going to be 9. correct nine plus nine eighteen and then radius is 12. so how do we find X okay I I don&#39;t access the height of the triangle okay so X is pythagore&#39;s theorem okay so x equals to square root 12 squared minus 9 squared you should get three set seven okay so I&#39;m going to leave it in this like I said I want to round round off the number so quickly so I&#39;m going to use three seven okay CM now you want to find the height right height is here so the height is equivalent to the radius which is 12 minus 3 7. so your answer is going to be 4.0627 cm okay this is the height next question they ask you to calculate cross-section area covered by water so basically they&#39;re asking for area of the segment correct you see where is the water in the diagram this part right so it&#39;s the area of the segment so how do you find the area of segment the formula correct format sorry question B formula is what area of segment one over two R square Theta minus sine Theta okay the first Theta is in terms of radian the second Theta is in terms of degree okay so now how are we going to find the angle huh so we know that we are looking for this angle right this whole angle is what the whole angle is Theta but now what I can do is I can use this triangle to form find this right and I times 2 I will get two sides correct so I&#39;m basically using the triangle so I can use one socatoa right so if this whole thing is pi sorry if this whole thing is Theta the half here is called Theta over 2 and here so Theta over 2 correct so let me redraw this so you got a triangle now case 12 he is 9 here is Theta over 2. okay because the whole thing is Theta so the half is only Theta O2 okay so now we can use socato so in this case what are we going to use opposite and hypotenuse so it&#39;s sine right so sine Theta over 2 equals to opposite is 9 over 12. okay so you can shift this you should get your Theta is equivalent to 97.18 degree so this is in degree but you also need in terms of Radiance times pi over 180. so you should get your answer 1.6963 radian okay so one in terms of degree one in terms of radian now you can find your area of segment one over two times R what&#39;s the radius radius is 12 12 square multiply by Theta this versatiles in terms of radian right so 1.6963 minus sign this is in terms of degree so 97.18 okay key this into your calculator you should get your answer straight away 50.7019 cm square okay so this is how you solve much challenging questions so I hope you guys understand this okay now let&#39;s see more examples so the diagram on the right shows two discs with radi 11 cm and 7 cm touching each other at point R okay that these are on a straight line p d c q calculate b a d in degree okay so let me redraw these are you got basically like a trapezium right so here 90 degree here 90 degree here is 7 cm here total is 11 cm okay so now they want you to find bad where&#39;s bad here is a CE B here is D here you see so bad should be the angle here okay so how in what way can you find that so I want your I want you all to understand this there&#39;s a trick usually right when there is a 90 degree there is always a right angle there is always a right angle okay so where is your right angle in this diagram the right angle is here you can draw a line parallel to the bottom line here okay so that means here is also 90 degree so now you have your right angle triangle okay you have your right angle triangle so if this is a right angle triangle we know this is 7 cm because it&#39;s going to be the same as this right so what is the length here 11 minus 7 so here is 4. okay you want to find this angle you know these four what other data can you get you know you know this length you should know this length how to know because you notice that in this diagram they are the radius correct this radius here is 11 and then this radius here is seven so that means this length a b is actually 11 plus 7 which is 18. okay so now you have this length you have this you have this so you can use your soccer okay to find the angle uh bad so question a you want to find angle BD so what what are you going to use you&#39;re going to use um a Json and hypotenuse so it&#39;s cos okay so cos angle b a d equals to adjacent is 4 hypotenuse is 18. okay so what&#39;s your angle bad should be equals to 77.16 degree okay so they want in degree right so this answer question B subsequently find the Shaded area so shaded area is where actually if you don&#39;t didn&#39;t notice this is the Shaded area okay this is the Shaded area so how are you going to find this area the only way you can find this area right is by using the whole trapezium here find the total area minus this area the area of this sector and minus the area of the sector okay that&#39;s it when you can do that you get your answer shaded area so area of trapezium you can find because you know the formula is what one over two the two parallel sides you add them up a plus b times the height okay so the height is here but you do know what is the height correct so how are you going to find the height the height is going to be you can use back this triangle right you know this and you know this so to find the length CD you just have to do Pythagoras Theorem okay to find okay let&#39;s do it together so question B okay I&#39;m just gonna draw here so there you go space question B so let us first find the height here okay so let&#39;s assume this is the height so height is equals to 18 square minus 4 Square and then square root okay so your answer should be 2 3 77 okay I&#39;m gonna leave it in this form I don&#39;t want to round off first if I write in decimal I have to round off so I don&#39;t want that to do that so early in the process so this is 277 cm so you can now find the area of the entire uh trapezium so now you have to find area of the SEC sector yeah so this sector you can find because you already know this first sector here this one you can find because you know the radius because the formula is what 1 over 2 R square Theta right so Theta is the angle radius R is radius so you already know the radius and you know the ta the angle which you already found here in the 77.16 but you have to change to radian later the second sector you have to find is this way right but this one is a bit hard because first thing is you don&#39;t know uh X actually you already know the radius which is 7 cm but you don&#39;t know the angle right you don&#39;t know what&#39;s the Theta so now you have to find the Theta first so how are you going to find the Theta how are you going to find so the only way you can find is actually maybe there are other ways left but the only the way that I can think of is we can take you know that a full trapezium What&#39;s the total angle 360 right here is 90 here is 90. here your default is 77.16 so now you can find this right correct so you can just straight away do that to find the angle so we are going to do that okay so let me just erase this okay all right so let&#39;s find uh angle um angle a b c okay so that angle is going to be equals to 360. minus 90. minus 90 minus 77.16 so that angle is equals to 1 0 2.84 degree okay so this angle of course uh we have to change to radian because we want to use to solve the area okay so now we&#39;re going to straight away do that area of shaded region okay area of Shield region is equals to you take the whole area right area of the trapezium so one over two times a plus b what is a and what is b a is seven the two parallel sides are please don&#39;t take this and these are Noah must be the parallel sides so the parallel sides are seven and the other side is 11 so 7 plus 11. multiply by height is what you recalculated 277 okay minus area of the first sector there which is 1 over 2 R square is R is 11 squared Theta is what Theta is bad right so it&#39;s 77 .16 don&#39;t forget to change to radian so pi over 180. and then minus the second sector 1 over 2 R square R is seven square and then the angle is um one zero two point eight four so one zero two point eight four times pi over 180. okay so use your calculator you should get 18 77 here minus 81.4857 minus 43.9807 so your final answer should be 32.4830 cm square okay so that&#39;s how you solve this kind of question all right next the diagram on the right shows a moon shaped kite whose line of symmetry is OS okay for this question right I really need you all to see closely because it&#39;s quite tricky okay so just look at this huh so aqb where is aqb this aqb right aqb is Arc sector Arc of a sector from from a circle with Center also sorry so here is a sector okay I need your take note because if you don&#39;t see closely which Arc it belongs to and you will get you will in the end right you don&#39;t know which radius to use okay so this question is a bit Messina so I need you all to take note of that so now in this case this Arc what is the radius radius is 20. the center is O Okay so take note of that the second one they say is what a p b r a base a p b r a p b r is a what semicircle so that means for this Arc this Arc here if you want to find this Arc you need the radius is 16 okay please don&#39;t use the wrong radius if you use 20 you get the wrong answer so okay so that&#39;s done next they tell you radius 16 Tru is an arc from Circle with Center as so this one is quite straightforward this arcade belongs to this sector okay and then the radius is 12 and then given that Arc Length of Tru so this outline here is 21 cm calculate aob and TSU where is aob aob is here TSU is here so TSU is actually quite easy right this is quite easy okay let&#39;s find TSU first huh so TSU you can just use the formula s equals to R Theta so the S is 21 R is 12 Theta is Theta so theta equals to 21 over 12. which is 3 oh sorry 704 so right here seven over four okay I try not to simplify this so that um in the end later um anyway this one I think you can do it with this one you can simplify to 1.758 so I just gonna write 1.75 1.75 radian if it&#39;s a recurring decimal I try not to simplify so soon okay because otherwise the final answer will be will be very far from the actual answer so now this is Tsui angle t s u now you want to find angle aob so angle aob let&#39;s say this Theta okay so this you know this is 20 this is 16 so you can form your right angle here right so this small angle here is Theta over 2 okay because it&#39;s you cut into half so Theta over 2 is that&#39;s half of that angle so we&#39;re going to use um opposite this angle here is of this length is opposite this hypotenuse right so opposite hypotenuse is sine so sine um Pi will give it sine Theta over 2 okay I cannot use Theta because you overlapped with the this previous question here this one so I&#39;m not going to use Theta let&#39;s just use angle aob over 2 okay sine of this is equals to opposite which is 16 over adjacent so you shift one by one you should get angle A or B equals to 1 0 6.26 degree make sure you change to radian because it asks for Radian times pi over 180 you will get 1.8548 Radian okay so this is um the two angles that you&#39;re looking for okay next they ask for perimeter so where&#39;s the parameter perimeter here you already know 21 here you also know 12 and 12. so all you have to find is this and this okay when you can find the two then you can find the perimeter so let&#39;s find um question Bia let&#39;s first find ARB so ARB is equals to so we use this as a semicircular at least so I say you have to know which radius to use okay so since it&#39;s a semicircle is going to be um R Theta right R is 16. Theta is the angle is half a circle so I told you 480 is equals to PI right in radian form right so this is pi so you get 16 pi I&#39;m just going to write 16 pieces okay CM after that you&#39;re gonna find for aqb aqb is using the sector okay so the sector the radius is 20. Theta is you found the aob right so it&#39;s this one so it&#39;s going to be 1.8548 so you get your answer is 37.0966 CM so you want to find perimeter right so I&#39;m going to write here perimeter equals to 37.0966 plus 16 pi plus um 12 plus 12 plus 21. okay your final answer should be one three two point three six eight six cm so now we have to find for the last question which is area so how do you find area so area here is a bit complicated so yeah okay let me erase this because I don&#39;t have enough space okay here is this and this one okay now we want to find for area this area is very easy to find right because you just use the formula however if you want to find this other shape here this moon shape is a bit hard because why first thing is there is no formula for this second thing is you have basically two shapes involved first is a semicircle and a sector so take note now how are we going to do this we are first going to take the semicircle yeah use a different color we take the semicircle now the semicircle you have to minus this correct to find this area so how do you find the area of that segment you can use the formula area of segment formula but in this case I&#39;m not going to do that what I&#39;m going to do is I&#39;m going to take the entire area of the sector entire area of the sector minus this triangle because we know it&#39;s a triangle right so I&#39;m just going to minus that also again okay you can use that or you can use the formula of the segment area of segment formula okay so I&#39;m going to use the way that I just mentioned minus a triangle so I mean let&#39;s do this together so Ariel question C here area is equals to let&#39;s find this first this one so it&#39;s very straightforward 1 over 2 r squared what is r 12 square um Theta is where is TSU okay here 1.7 5. plus okay now here is where it gets tricky so you want to First find the semicircle first right so semicircle is 1 over 2 r squared what is r r is 16. Square um Theta okay I don&#39;t know whether you can see or not because I color okay so are the radius is 16 and then Theta is one half a circle is pi okay minus so the semicircle must minus the sector correct but the sector is what how to find the sector the sector you have minus 1 over 2 um our R is 20. r squared theta wasted is um one point eight five four eight minus the triangle so triangle is one over two times base basis 16 plus 16 is 32 it is the beta okay or what about the height okay so the height here is op how to find op you can use this triangle okay so op is equals to 1. 20 squared minus 16 square a Pythagoras Theorem so you should get 12 okay so that is 12 CM so 1 over 2 times 32 times height is 12. then you close all the bracket okay I know it can be a bit confusing and I&#39;m sorry I&#39;m writing it this way because it&#39;s uh so hard to see but as long as you understand what I&#39;m trying to do okay so in the end you should get so this part here you should get it is the area of t r u s correct this sector here so that area here should be one two six Plus so this area of the semicircle area of semicircle versus semicircle a p b a p b r yeah apbr should be four zero two point one seven six okay minus minus what should be the sector the sector should be what is the sector again A O B Q right aob Q is 370. 96 minus the triangle which is 192. okay so in the end you can use your calculator you should get your answer is three four nine point two one six cm square oh my God I hope I hope it&#39;s not confusing okay okay I&#39;m gonna repeat this okay because I don&#39;t want you all to get confused let me erase all this okay so basically what is happening is you&#39;re gonna find this area which is this part okay area of the t r u s okay plus area of the semicircle semicircle is um AP B R okay minus you want to find the sector so how to sorry you want to find the segment so how to find the segment the segment is you have to first find the sector minus the triangle up here okay so the sector is um a o b q minus the triangle triangle aob AO B pill okay so basically this is what I&#39;m doing here okay so the trus okay this sector is this one and then I Plus apbr which is the semicircle so this is apbr minus this one is to find the area of segment which is what I did here this part okay so I hope you find this uh more and able to understand okay so I hope I didn&#39;t confuse you guys all right let&#39;s see the next question we have two more questions and we&#39;re done with the chapter so let&#39;s see this in the diagram on the right are three identical 20 cent coins with the same red eye and touching each other if the blue colored region has an area of okay so we give you the area of the colored region find the radius of each coin okay so they give you nothing no data except the area of um the color region so how are you going to find so if you notice is a tricky question but what you can do is here there&#39;s a center here correct there&#39;s a center here for each of the coin what you can do is you can connect the radius here is the radius for this coin and here&#39;s the radius for this coin here&#39;s the radius for this coin [Music] and here&#39;s the radius for this coin the radius here and the radius here Okay so here is R here is R here&#39;s R here&#39;s our his R user because all same radius right same coin same radius so in other words you have a triangle okay you have a triangle here is 2R here is 2R here is 2R okay so since they are all same length or two are right so it&#39;s an equilateral triangle and for an equilater equilateral triangle what are the properties so the property is all three angles are the which is 60 degree okay 180 day battery so you get 60 degree each of the angle so now we can use this info to solve okay so what is the formula we want we can to find the radius right we have to form an equation using this okay because only this is given to you so how to find the area of colored region how do you find that area the area of the color region is equal to area of the triangle minus three times of the area of sector okay so in other words you just have to find each sector okay if we just find one sector times three you&#39;ll get three sectors so the whole area of triangle minus the three sectors you should get the area of the colored region so let&#39;s form the equation okay so color region is given to you 12.842 area of triangle so you can use 102 times base times height but in this case the easier one would be we learn from four area of triangle one over two a b sine C okay remember I think I told you all Chopstick right like Chopstick so a b sine C so the two length if let&#39;s say I&#39;m using this angle right so the two length must be this two length okay if I&#39;m using this angle the two length should be these two length okay so 1 over 2 a B is 2R times two r sine C so a 60 degree so make sure you 60 degree because this one is sine cos tangent remember I said sine cos tangent you have to use in degree okay if it&#39;s uh the normal formula for this chapter then you use in radian so in this case this is the formula minus 3 times area of the sector so 1 over 2 1 over 2 times R is what the radius is um each of these R right so it&#39;s r Square times um the angle is 60 degrees so 60 please change it to radian so pi over 180. okay so you will get 1 2 8.3 12.842 equals to third tree R square minus pi over 2 R square and then you rearrange this you should get R square equals to um 79.738 okay so in the end your R should be equals to 8.9297 [Music] this is what mm mm yeah okay so the actual answer that was given is actually eight point nine three one so it&#39;s actually pretty close so it&#39;s acceptable okay so it doesn&#39;t this kind of this chapter right it you won&#39;t get a accurate like exact answer because there are many decimal places involved okay some people might just take three decimal place and then the pi some people might use uh 3.142 some might use 22 over 7. in this case I&#39;m just using 3.142 okay uh some people might use the one in the calculator okay so you will in the end will get different answers okay yeah so this is how you solve this question don&#39;t be uh frightened by the question okay try to figure out how can you solve this because only one value is given so you must try to figure out how you can solve this okay all right one last question let&#39;s go [Music] the diagram on the right shows the logo of an ice cream company the logo is made up of three identical sector okay so these are the sectors here the three sectors okay from the center and radius of 30 cm so all these are the radius 30 cm30 same because all identical right and then it&#39;s given that all these angles are 60 so here 60. here&#39;s 60 calculate equation a they ask you to calculate up length of a b raise a b a b is here so what&#39;s the formula s equals to R Theta so I&#39;m just going to write a b a b equals to radius is 30. Theta is 60. please change it to Radian okay so you should get your answer 3 1.42 cm okay that&#39;s the first one second okay I need to cram a bit because I don&#39;t think we got space here but I mean um second one is area of cod right cod cod is here they&#39;re asking for sector right so you have to include the segment as well so this whole thing okay so we just use the formula of sector so area of sector Cod is equals to 1 over 2 R square what is r 30 square and then Theta is 60 times pi over 180. so your answer should be 150 Pi also equals to 471 .3 cm square okay so if you get anywhere close to 471.15 is acceptable okay that&#39;s the actual answer next you want to find for perimeter of segment EF where is EF okay EF is here so this is a perimeter so the outline you already found right outline is this because this identical shape so here is 31 point 31.42 yes so now you want to find the EF this straight line here how to find that so what you can do is you can form your try you can see your triangle right your triangle oops sorry okay so this is e f so you can form your triangle here 90 degree okay so what you can do is here&#39;s 60 degree right so this one is going to be 30. 30 degree you know the radius is 30 cm you want to find here okay so what is that angle that let&#39;s say this is um opposite right opposite so opposite and hypotenuse so you can use sokatoa so you can use a o o and H so sine so question three sine 30 is equivalent to opposite over 30. so what&#39;s your opposite angle opposite angle should be equivalent to 15 CM okay so here is 15. so now you need to find 15 and 15. here let me just right here so this half here is 15. here is 15 years of 15. right and then you want to find the parameter okay so perimeter is equivalent to 31.42 plus 15 plus 15. so your final answer is 61.42 cm okay next area of segment EF so this one I already gave you all the formula right just straight away use the formula no need to thinkery so area is equivalent to 1 over 2 r squared what is r the T Square Times Theta is 1 Theta is 60 so 6 you have to change to a radian 60 times pi over 180 minus sine 60 okay because this must be in 3 and this one must be in degree so you should get straight away you can get your answer 81.5886 okay now like I said it doesn&#39;t have to be 100 accurate okay as long as it&#39;s close to the answer because you might use different value of pi okay or you might use a three decimal place or two decimal place okay so in the end you&#39;ll get a you may not get the exact same answer but if it&#39;s close it&#39;s acceptable okay last two question um the logo is casted in cement if the thickness is uniform and it&#39;s 5 cm so the thickness now is 5 cm okay find the amount of cement needed in to make the logo so to do that you must first find the total area and then times the thickness okay the total area that means in 2D form of the logo times the thickness so what is the total area actually yesterday we find the total volume so total they ask for volume right equals to what&#39;s the area area of the whole sector right so you got Cod is this area one area right of the sector so you take three times of that which is three times four seven one point three and then you times 5 cm because that&#39;s the thickness so in the end you should get seven zero six nine point five is a volume okay so that&#39;s how you find the volume last question question C what are they asking if the cost of cement is 50 cents per CM Cube find the total costs okay so they are saying one cm Cube is 50 Cents right so how many you have how much you have you have seven thousand and 69.5 cmq is equivalent to this multiplier okay actually I shouldn&#39;t write equal I should just okay this one is 7069.5 times 0.50 so your answer should be RM three five three four point seven five okay so this is how you solve this kind of question it&#39;s actually pretty straightforward write this question it looks long but it&#39;s actually very straightforward all right guys so that&#39;s all for this video These are the four main things that we have covered in this video we have covered the whole entire chapter we have covered radian Arc Length Arc of the sector and we also covered out of segment and then some application questions all right so that is all for this video I hope you find this useful if you did don&#39;t forget to like subscribe and share it with someone who will also find this beneficial right so until next time hope to see you on my next video alright take care bye

Transcript for:Understanding Circular Measure in Mathematics

Transcript for:
Understanding Circular Measure in Mathematics