now let's talk about arc length and how to find it so let's say if we have a circle and we only want to find a portion of the arc length of the circle let's say this portion the length of that segment in green is known as s which is the arcl length and it's based on the angle Theta let's say C is the center of the circle and this is a and b the distance between points A and B around that circular around the edge of the circle is s s is the arcl length the distance between C and B is the radius which is the same as from C and A so the Arc Length s is equal to the angle in radians times the radius so for example let's say if we have an angle of five radians and a radius of 12T what is the arc length so imagine a picture where R is 12T and the angles five radians so this is about five radians keep in mind 6.28 radians is a full circle 3.14 radians is half a circle so five radians is more than half a circle so we want to find the distance between these two points around this portion of the circle that's what we're looking for so s is equal to Theta * R the angle is 5 radians and the radius is 12T but I like to think of it as 12T per radian when I think of the word radius I think of the length per radian and notice that the unit radians cancel so 5 * 12 is 60 so the arc length is 6 60 fet that's how long it is let's try another example so let's let's say if the radius is 8 cm and the angle Theta is given to you in degrees let's say it's 150° what is the arc length so if you want to draw a picture this angle is about 150° so this is8 that's eight and we're looking for this distance here and here's the angle so go ahead and calculate s now what we need to do is convert Theta into from degrees to radians to do that let's multiply it by Pi / 180 so we can cancel a zero and so we have 15 over 18 15 is 5 * 3 18 is 6 * 3 so this equation to 5 piun / 6 now to find the AR length it's Theta * R so it's the angle and Radiance 5 pi/ 6times the radius of 8 cm per radian so this is going to be 40 Pi / 6 which we can reduce it to 20 pi over 3 so this is the answer that's the arcl length now if you want to get the decimal value of it this is going to be about 20944 CM now if you don't want to use this equation there's another equation that you could use if you want to keep your angle in degrees so the arc length is equal to the angle in degrees / 360 times 2 pi r where 2 pi r is the circumference of the circle so if you think about it let's say if we have a circle the circumference is 2i r a full circle has an angle 360 so if you plug in 360° it will be 360 over 360 which is 1 * 2i R and the r CL will simply be 2 pi r now let's say if you have a fraction of a circle like half a circle the angle is 180° so it's going to be 180 over 360 * 2i R so it's basically 12 of the entire circle so the circumference of a semicircle is pi r so as you can see you could think of the arc length as being the circumference times a fraction of a circle and that's what Theta over 360 is it's a fr fraction of the circle so if the angle is 90 that means what we have is 1/4 of a circle so this is an angle of 90° as you can see the arff is going to be 1/4 of the circumference or 1/4 of 2i R but now let's go ahead and finish this problem so this is going to be 150° / 360° * 2 piun * the radius which is eight so we can cancel a zero I would start with that now 15 I'm going to write it as 5 * 3 36 is 3 * 3 12 but 12 I'm going to break that into 4 * 3 and 8 I want to write that as 4 * 2 just to do some canceling so I can cancel a three and I can cancel a four and that's about it so now let's multiply so 5 * 2 is 10 * 2 is 20 so I have 20 pi and on the bottom a 3 now Pi is about 3.14159 If I multiply that by 20 and divide it by 3 I'm going to get the same answer 20.9 44 and that's the arc length in centimeters so there's two equations you need for arft S equal Theta R and only use that if Theta is in radians now you can also use this equation Theta / 360 * 2i R where Theta is in degrees so those are the two formulas that will help you to calculate the arc length of a circle now for those of you who want access to my complete online trigonometry course here's where you could find it uh go to emi.com and then in the search box you could just search for trigonometry and you can see my course is basically the one with the black uh background and then here is it I'm still adding more lectures but here's what I have so far um introduction into angles drawn angles converting degrees into radians uh linear speed angular speed problems Arc Length uh information on the unit circle how to evaluate trigonometric functions using the unit circle right triangle trigonometry things like sooa even you could have video quizzes as well solving work problems like angle of elevation problems and then you have the next section graphing s cosine functions secant tangent inverse trig functions pretty much all the common stuff that you'll see in a typical uh trigonometry course even solving uh barings verifying trigonometric identities summon difference formulas double angle half angle and some other things too and as I mentioned before I'm going to add some other things as well so feel free to check it out when you get a chance and uh let's continue back to the video let's say if we have a circle and we want to find the area of a fraction of that Circle let's say this is 90° and we're given a radius of 10 cm what is the area of the Shader region how can we find the area of that sector of the circle well you need to use this equation 12 Theta R 2 but this equation only works IFA is in radians now if you multiply that equation by 2 Pi / 360 this will convert the angle from radians to degrees so notice that the twos will cancel and so the area is going to be Theta over 36 * PK R 2 so Nowa is the angle in degrees and if you think about it PK r s is the area of the entire circle and Theta over 360 is the fractional part which represents the area of the sector relative to the entire area of the circle so in our example the angle is 90 and radius is 10 so P pi r 2 the area of the entire circle is 100 Pi 90 out of 360 is basically 1/4 so to find the area of that sector it's 1/4 of the area of the entire circle so 1/4 of 100 Pi is 25 pi and that's how this equation works it's simply the fractional part of the entire circle so Theta over 360 is just the fraction of the circle that represents that small sector times the area of the entire circle so make sure you're aware of these two equations 12 Theta R 2 that's for the angle if it's in radians and Theta / 360 * < r^ 2 if you have the angle in degrees now if you recall going back to AR length we said that the arc length is equal to Theta * r where Theta is in radians now to get the other equation you can multiply it by 2 Pi / 360 and if you rearrange a few things this is equal to Theta over 360 * 2i R so notice how this equation is similar to this one the area of a sector of a circle is Theta over 360 that's the fractional part times the area of the entire circle PK r² the arc length is the fractional part of the circle which is Theta over 360 times the circumference 2i R and you can get it by multiplying the original equation by 2 pi over 360 the same way in which we got this equation from this one where a was equal to 12 Theta R 2 where Theta was in radians all we need to do is multiply by 2 pi over 360 and we can get this equation so I wanted to highlight the similarities between these two processes now let's say if we have a circle and we want to find the area of this sector let's say the angle is 60° and the radius is 5T go ahead and find the area of the Shader region so because the angle is in degrees let's use this equation th / 360 * pi r 2 so it's 60 over 360 * piun I put equals for some reason but that should be a multiplication symbol so time Pi * 5^ 2 so we can cancel with zero 6 / 36 reduces to 1 6 and 5^ 2 is 25 so the answer is 25 piun / 6 and if you want a decimal value for that multiply 25 over 6 by 3.1459 and you should get about 13.1 square feet because we're dealing with area area has the units square units in this K Square ft let's try another example so let's say the angle is two radians and the radius is 8 cm that's the distance between these two points calculate the area of the Shader region find the area of that that sector so in this case we want to use this equation area is equal to 12a R 2 because the angle is in radians so the angle is two radians and R is 8 cm so 12 * 2 is 1 and 8 s is 64 so the area is 64 square cm