[Music] oh [Music] okay welcome to this video where we're going to have a look at all the formulas that you need to get a pass in your gcse maths exam now before we get started these are going to be the crossover formulas so these are the formulas that you need that appear in both the foundation and the higher paper if you want to look specifically just at the higher topics and just the formulas that you need in the higher tier paper then do make sure you check out the next video in the series but even if you are doing the higher tier paper you are going to still need to know all these formulas so i'm the gcse math tutor let's have a look at all of these so the first one that we're going to have a look at here is going to be the area of a rectangle so the area of a rectangle is length times width now when you have a question like this and for all of these questions they're going to be relatively simple examples but if we have a rectangle and we have the lengths 3 and 8 they are the length and the width so we would just do three times eight and that would give us an answer of 24 and not for getting our units there centimeter squared okay on to our next formula so we'll now have a look at the area of a triangle the area of a triangle is half base times height and when these formulas are written they're not they don't tend to be written with the multiplication symbols but i'm going to keep them in there for the purpose of this video so if we look at a basic example if we have a triangle and we're given the lengths when it comes to using this particular formula we do need to know the perpendicular height so if we have the perpendicular height we can use this formula and we just need to make sure we identify the correct lengths so the base here would be 10 and the perpendicular height would be 7 so we would do half times 10 times 7 and that gives us an area of 35 centimeters squared what you can do is you can just halve the base number so that would be half of 10 which is 5 and then multiply it by the height which again gives you 35. okay so on to our next one we'll look at the area of trapezium now the area of the trapezium is half times a plus b times by the height now a plus b the a and the b when it comes to a trapezium are the two parallel lengths and in this particular trapezium that's the top and the bottom and they would normally be labels a and b so for this one here we need to actually add a and b together and we also need to know again the perpendicular height so if we add in the perpendicular height and we'll say that that is five for this one here we would do a half times a plus b which is eight plus ten and then multiply it by the height which is five we can simplify that a little bit further because we can add the 8 and the 10 and that would give us 18. so if we did a half times 18 times 5 that would give us 45 centimeters squared and you can take a similar approach to that we did with the triangle you could halve the 18 to get 9 and then multiply it by 5. so there's the area of a trapezium let's have a look at our next one which is going to be the area of a parallelogram so for the area of a parallelogram this is quite a nice one very similar to that of a rectangle it is base times height so when we're looking at a parallelogram we obviously need a base and a height and again we need to remember that the height there is the perpendicular height so we'll have to add that in if we can actually work out the area of the parallelogram so for this one here the base is the same as the top so we could move the top down to the base and again now we would just need to do eight times five which would give us an area of 40 centimeters squared okay on to our next one we're gonna have a look at the area of a circle so the area of a circle gets slightly more complicated as we start to introduce pi so you need to know how to find pi on your calculator but the formula is going to be pi r squared or that can be written as pi times the radius squared again they mean the same thing but just introducing that multiplication symbol so again if we take a circle we do need to know some of the parts of the circle we need to know that the length all the way across is called the diameter and we also need to know that half of that distance is called the radius and for finding the area we're going to be using the radius so for one of these questions you might be given the diameter and have to halve it to find the radius but for the purpose of this example we're just going to put in a radius of 4. so if you were to type this into your calculator you would type in pi times 4 squared and that particular multiplication here gives us quite a long decimal so you need to read the question carefully but for the purpose of all of these examples we're just going to round the answer to one decimal place again making sure in an exam though you actually write down all of the numbers unless you are specifically asked around it but let's imagine we were asked to round to one decimal place we would round that and say 50.3 centimeters squared now sticking with the theme of circles let's have a look at the circumference of a circle so for the circumference of a circle we have another formula and that is pi times the diameter or you could also say pi times two of the radiuses or pi times 2r so if we look at a circle first of all we need to know the circumference is that distance all the way around the outside and in order to find that we just need to plug our values into our formula so for this particular one we're going to be using the diameter but again you might be having or to use the radius and doubling it to get the diameter so let's say the diameter is six centimeters and then plug that into our formula which is pi times six again this gives us quite a long decimal so if we have to round it again i'm going to round this to one decimal place but just making sure that you read the question and round it as you have been asked to so for this one here that would be 18.8 and in this case it's centimeters as we are not looking at an area this time all right we're going to move on to some volume so for the volume of any prism the formula is to work out the area of the cross-section and then multiply it by the length so for this particular example we're going to have a look at a triangular prism now we know if we are looking at the cross section we are looking at the face which runs all the way through it and in this case this is a triangle and we've already looked at how to work out the area of a triangle so we'll have a look at that in a second the length is the distance that it goes through the shape so in this case it would be that length shown for you so if we add in some lengths we can actually go about working out the cross section for this particular one we are going to need to know the perpendicular height of the triangle as we discussed earlier so let's add in a perpendicular height of four again potentially you could have to use some other mathematics to work that out but for keeping simplicity for these examples we'll just chuck that length in so for this one for the area of the cross section we would do a half times base times height for the triangle and that would be a half times six times the perpendicular height of four which would give us an area of twelve centimeters squared we then take that area and multiply it by the length the length here is 10 so we would just do 12 times 10 which would give us 120 and this time the units a centimeter cubed as we are looking at a volume okay let's have a look at another volume and that is the volume of a cylinder so the volume of a cylinder is worked out in exactly the same way as if we as what we actually have is a circular faced prism but in order to work out the circular face we need to do pi r squared so our formula here is pi r squared multiplied by the height so if we have a look at our cylinder and think about the fact that we do have these circular faces we obviously need to know a few elements in regards to the cylinder we need to know the radius of the circle so let's put in a radius of 3 and let's put in a height of eight so in order to work out this volume first we would have to do pi r squared so pi times three squared would give us this long decimal so again as we are now going to work out the volume you definitely do not want to round your working out so we'll take that decimal there in our calculator and just multiply it straight by the height which is eight so i would press times eight and we would get and again we get another long decimal here again just reading the question and making sure you round that as you have been asked but if we stick to the rest of our rounding that we've done in this video we'll round it to one decimal place and we'll get an answer of 226.2 centimeter cubed okay moving on to our next formula we're going to have a look at pythagoras theorem now pythagoras theorem is a squared plus b squared equals c squared and this applies when we're looking at certain types of triangles it specifically applies when we're looking at a right angled triangle so we will have to have a right angle in our triangle in order to use this but our lengths here would be a and b now the a and the b are interchangeable so the a could be b and the b could be a in this example but i've just decided to label it that way now the hypotenuse which is the longest length is what we call c in our formula so these are the lengths that we're going to be looking at when using pythagoras theorem now if we replace those with some numbers let's say that we have three and four and we're looking for the hypotenuse so a and b would be three and four so we'd put those into our formula and we would have that three squared plus four squared is equal to that c squared if we simplify three squared plus four squared we get the answer 25 so 25 is equal to c squared and if we want to know what c is we would have to square root 25 which would give us the answer 5. remembering that when you square root a number like 25 you do actually get two answers plus and minus five but obviously we're looking at a length here and a length cannot be negative so the answer must be 5. when you are looking at pythagoras you could also be looking for one of the shorter lengths in which case you actually have to do a subtraction maybe it's c squared take away a squared which will equal b squared but you need to know this formula and then looking at rearranging it if needed now moving on from pythagoras we can also look at right angle triangles when it comes to our trigonometric ratios and again this is for right angle triangles but this is when we have an angle involved so if we put an angle into this triangle and let's call that angle 30 degrees for the purpose of this example then we need to know how to label the sides so the side that is opposite the angle if you were to draw an arrow out of it is called the opposite side the next side which we already know from pythagoras is the hypotenuse which is the longest length which is always opposite the right angle but don't get that confused with the opposite side as we are looking at the angle that's not the right angle we also then have the final side which is called the adjacent side so these are the three lengths that you need to know when using any trigonometric ratios now there are three formula triangles are very helpful for this and that is shown on the screen so we have that sine is equal to opposite over hypotenuse cos is equal to adjacent over hypotenuse and tan is equal to opposite over adjacent now if we focus on just using one of these for our example we'll look at using our sine triangle so if we get rid of these labels and we actually put in some numbers let's say that this side is 5 and we're looking for the hypotenuse now in this case we are given the opposite which is 5 and we're looking for the hypotenuse and o and h are in our sine triangle so if you cover up the hypotenuse which we're looking for it tells you it's o over s so the hypotenuse is o divided by s and in this case that would be 5 divided by sine 30. so if you type that into your calculator you actually get quite a nice number for this one you get the answer 10. normally you're going to get decimals here but there is something else involved and that is that the value of sine 30 actually has an exact value of one-half and this is just one of the trigonometric exact values that you do need to remember and again don't forget i'm going to link any videos to all of these in the description if you need to have a look at these a little bit further so let's get rid of that and let's maybe have a think about just a slightly different example so instead of using the sign let's use the cos triangle and let's instead put some different numbers in and we'll say that the hypotenuse is 8 and this time we'll look for the adjacent side now the adjacent if you cover it up in your triangle is c times h so a would be equal to cos with the angle of 30 multiplied by eight and again if you type that into your calculator this time you do get a decimal and again if we rounded that to one decimal place we would have 6.9 centimeters so there is a few examples but also you need to know how to work out an angle and how to use the inverse of these if again you are looking for the angle rather than the lengths so again don't make don't forget to check out the video if you're not sure on how to do some of those so moving on to some of our final formulas we're going to have a look at our density formula and density is equal to mass divided by volume again you can put this into a formula triangle and we can have a look at an example question so it's a nice simple one but if we look at the mass of a solid being 200 grams and the volume of the solid being 40 centimeter cubed if we were to calculate the density we would just plug that into our formula mass divided by volume would be 200 divided by 40. and that comes out as an answer of five we just need to get the units correct here and you can pinch the units from the question we had grams and centimeter cubed so the units would be grams per centimeter cubed so that's for density let's move on to speed and speed is going to be equal to distance divided by time again that can go into a formula triangle and you can use that as you see fit but we'll have a look at an example so for this one we'll say a car travels 448 miles and it takes the car eight hours calculate the average speed so again plugging this in we would do 448 miles divided by eight and when we work that out it gives us an answer of 56 and when it comes to speed again we were looking at miles and hours so we would give that speed as being miles per hour and there we go again you could obviously rearrange that with your triangle for working out this distance or time so on to our final one we'll have a look at pressure and pressure is equal to force divided by area so again this can go into a formula triangle and let's have a look at a final example so it says here that a solid exerts a force of 3500 newtons on the floor the area of the solid in contact with the floor is five meters squared calculate the pressure exerted by the solid on the floor so for this question here again we would plug in the values 3500 divided by five that equals 700 and for this particular question we were using newtons and meter squared so we would you didn't use a unit of newtons per meter squared so there we go that's all the formulas just to obviously make you aware again these are the formulas that you need whether you are doing higher or foundation if you are doing the higher exam there are more formulas that you need and i will put the next video here in the description so that you can have a look at the higher only formulas as well i hope you found that video useful and helpful if you did please don't forget to like this video please don't forget to share it and do not forget to subscribe to the channel but until the next video i will see you later [Music] oh [Music]