Volume and Surface Area of Pyramids GCSE Maths

pyramid is formed by taking two-dimensional base like this quadrilateral here and then joining lines from each of the corners of this shape to point above the base like this it doesn't matter what shape the base is as long as it's polygon so it could be triangle like this or even hexagon like this all of these are examples of pyramids in your exam you'll almost certainly come across square base pyramid though and in this video we're going to learn how to find the volume and surface area of one now good news if your exam board is AQA they state that the formula for the volume in surface area of sphere cone and pyramid will be given in the relevant question so you don't need to memorize these formulas however this is not the case with other exam boards so you may need to learn this formula the formula for the volume of pyramid is 1/3 multipli the base area multiplied by the perpendicular height where the base area is just the area of the base and the perpendicular height is the vertical distance from the top of the pyramid all the way down to the base so that it forms right angle with the base you shouldn't confuse this length here as height because it's not perpendicular it's on slope so let's take pyramid and add some dimensions and work out its volume so we'll use the formula the volume equals 1/3 multipli the area of the base and for this one the base is square which is 6X 6 so its area must be 6 * 6 and then multiplied by the perpendicular height which for this one we can see is 5 so multiplied by five so we just need to work this calculation out now this one could actually be on non-calculator paper so we'd start by saying the volume equals and then we can multiply 1/3 by 6 by finding 1/3 of 6 and you do this by dividing 6 by 3 6 / 3 is 2 so this part here is 2 and then we need to multiply by 6 and then 5 2 * 6 is 12 and 12 * 5 is 60 so the volume would be 60 the units of volume are something cubed and since the lengths are measured in centimeters it would be cm cubed let's try second example so for this one we have square base once again with 6.5 and this perpendicular height is 11 this time you would be allowed calculator so using the formula the volume equals 1/3 multiplied by the base area so the area of this Square here 6.5 * 6.5 and then multiplied by the perpendicular height which is this 11 so multiplied by 11 so now you would just take your calculator and type this in exactly as it's written here that would give you this number and let's say the question said to round it to to one decimal place so it would be 154.32 type into the calculator and we'll get this answer here and it says to give it to one decimal place so it'll be 69.7 cm cubed for the second one we would do volume equals 1/3 multipli by the base area so 5 * 5 this time then multiplied by the perpendicular height which is 9 so * 9 now this one is non-calculator which you might find bit strange at first especially if you try to do 1/3 multip by 5 that's not going to give us an integer answer because five is in the three times table but since we just have lots of multiplications here we can change the order and actually do 1/3 * 9 first 9 is in the three times table so to find 1/3 of it you just divide 9 by 3 which is three so this part would be volume equals 3 and then multiplied by 5 ultip 5 3 * 5 is 15 and that multiplied by 5 is 75 so the answer is 75 cm cubed now sometimes we get trickier questions where we need to work backwards take this pyramid here and the question could say the volume of the pyramid is 320 cm cubed work out the perpendicular height of the pyramid so this time we've been told the volume and we need to work out one of the missing lengths let's imagine we were going to calculate the volume we would start by saying 1/3 multipli by the base area so 10 * 10 and then we would normally multiply by the perpendicular height but we don't have that it's called so we'll just write multiplied by but we do know the answer to this calculation we told the volume of the pyramid in the question is 320 so this must equal 320 now we just have an equation to solve if we start by multiplying 10 and 10 together well that's 100 so we have 1/3 * 100 * which is A3 * 100h which we could write as 100h over 3 so we end up with this equation here you can multiply both sides by three on the left hand side the threes will cancel so we just have 100h and on the right hand side 320 * by 3 is 960 then you can divide both sides by 100 100h ID by 100 is just 1 and 960 ID 100 is 9.6 so is 9.6 and the units would be cenm let's have look at another question with this sort of style so we have different pyramid this time and notice we're missing we told the volume once again but this time it's 735 and we need to work out which is the length of the side of the square base so let's once again imagine we were finding the volume we would do 1/3 multiplied by the base area but this time the base is square of length so * and then we would multiply this by the perpendicular height which we do know that's 20 and this must equal the volume given in the question 735 here we can multiply our and which will give you 2 so we have 1/3 * l^2 * 20 we can change the order of multiplications around so 1/3 * 20 * l^2 which is 1/3 * 20 2 or 20 l^2 over 3 so now we're going to solve this equation just like in the previous question we'll Begin by multiplying by three on the left hand side the threes will cancel so we have 20 l^2 and on the right hand side 735 * 3 is 2205 then we would divide both sides by 20 that will cancel the 20 on the left hand side so we just have and on the right hand side 225 ID 20 is 11.25 now this isn't the value of this is so we need to square root both sides of the equation if you square root you get and if you square root 11 10.25 you'll get 10.5 so the answer to this question is is 10.5 CM now let's have look at surface area the surface area of pyramid would be the total area of all of the faces square base Pyramid has five faces we have the square on the bottom this triangular face this triangular face this one and this one here so we have one square face and four triangular faces to find the surface area we need the area of all of these let's add some Dimensions to this shape let's say the length for the square base is 16 cm and normally we would draw on the perpendicular height if we were finding the volume this is actually not needed for the surface area instead we need the length for the side that goes from the top of the pyramid down to the midpoint of the base of the triangle so like this let's say for this one that length is 17 cm that will actually be the same length on all four of those triangular faces so this would also be 17 so with this and so with this as long as those lines go to the middle of the base of the triangle so to find the surface area we'll start with the area of this square base the area of the square base is just 16 * 16 or 256 now we'll take look at this triangular face here to find the area of triangle you do2 multiplied by its base so 16 multiplied by its perpendicular height which is 17 and this will give you 136 so the area of that triangular face is 136 but this triangular face is actually the same size and so is this one and so is this one so we could just take this area of 136 and multiply it by four to get the total area of all four of the Triangular faces which would be 544 so we could then take the area of the square base 256 and add to this the area of all four of those triangles which was 544 this will give you total surface area of 800 the units will be cm squared this time because we're talking about an area let's try second example so for this pyramid here we'll start with the area of the square base which will be 3.2 * 3.2 which gives you 10.24 then we'll look at one of the Triangular faces the area of the triangle will be2 multipli by the base so 3.2 multiplied by the perpendicular height 6.5 and this will give you 10.4 since there are four of these triangles we'll take 10.4 and multiply it by four which will give you 41.6 we can then add the area of the square base 10.24 and the area of all four triangles 41.6 and this will give you the total surface area of 51.84 CM squ here are two more pyramids for you to work out the surface area for the first one we'll start with the square square base so 6 * 6 which is 36 then we'll do one of the triangles which is 1 12 multip the base which is 6 multiplied by the height which is five and if you work this out you end up with 15 since there are four of those triangles we'll multiply the 15 by four which is 60 then if we add the 36 from the square base to the 60 which is the four triangles we get total of 96 CM for the second one we'll start with the square base once again 14 * 14 which is 196 then we'll look at one of the Triangular faces 12 * 14 * 20 and this gives you 140 there are four of those so we'll multiply 140 by 4 which will give you 560 and then we'll add the area of the square base 196 and the area of all four triangles 560 which will give you total of 756 and now let's have look at one final pyramid this pyramid is little bit different its base is no longer square but rectangle let's have look at how this affects our calculations for the volume and surface area we'll begin with the volume the formula for the volume still applies 1/3 multiplied by the base area multiplied by the perpendicular height so we would do volume equals 1/3 multiplied by the base area and the base is rectangle this time which is 50 by 22 so its area is 50 * by 22 and then we multiply by the perpendicular height which is 60 if you work this out on the calculator you'll get the answer 22,000 cm cubed so for the volume there's no major change we just need to do the area of rectangle for the base rather than the area of square let's have look at surface area so for the surface area we don't need the perpendicular height but we do need the lengths from the top of the pyramid to the middle of the sides of the base so this length here and also this length here and notice they're different this time because those triangular faces are not the same size we'll start with the area of the rectangular base which is 50 * 22 which gives you 1,100 then we'll look at this triangular face here this is 1/2 multiplied by the base 22 multipli by its height 65 and this will give you 715 the Triangular face that's opposite this one is also the same size but the other two faces are not the same so we only have two of these red triangles so we'll multiply 7155 by two which will give you 1 430 so now let's look at the other two triangular faces we'll do this one here which is2 multipli by its base 50 multipli by its height 61 and that gives you 1,525 but that triangular face is the same as the one at the back of the pyramid here so we'll also double this number so 1525 * 2 which is 350 so we now have the area of the rectangular base 1,100 the two triangular faces at the so 1,430 and the Triangular faces at the front and the back 3,50 so if we add all of this together we get the answer 5580 which would be the total surface area thank you for watching this video hope you found it useful check out the one think you should watch next subscribe so you don't miss out on future videos and why not try the exam questions in this video's description