In this video we're going to look at how to find the volume of a frustum, which is this odd looking shape. The key to this is to think of the frustum as a cone that's had its top bit chopped off. So a frustum is basically just a large cone that's had a smaller cone at the top taken away from it. This means that to find the volume of a frustum all you have to do is find the volume of the imaginary larger cone first.
and then subtract the volume of the imaginary smaller cone. The formula for the volume of a cone is 1 third pi r squared h, but to distinguish between the larger and smaller cones, we often use uppercase R and H in the formula for the large cone, and lowercase R and H in the formula for the smaller one. To see this in practice, let's give our frustum some numbers and try to calculate its volume. So looking at our formulas, the variables that we need to find are the radius and height of the large cone, and the radius and height of the small cone. We can tell straight away that for the large cone the height is 50cm and the radius is 10cm, because we were given those on our diagram of the frustum.
The small cone is a bit trickier though, because we're not directly given its radius or its height in our diagram. We can work out the height fairly easily though, because it's just this distance here, which must be the difference between 30cm and 50cm, so it must be 20cm. Now, to work out the radius of the small cone, we have to think of the large and small cones as similar shapes, and find out the scale factor between them. Because if we know how many times smaller the small cone is than the larger one, We'll also know how many times smaller the radius is. To find the scale factor, we can do the large cone's height, of 50cm, divided by the small cone's height of 20cm, which gives us 2.5.
So we now know that the small cone is 2.5 times smaller than the large cone. And so to work out the smaller cone's radius, we just do the radius of the big cone, the 10cm, divided by the scale factor of 2.5, which gives us 4cm as the radius of the small cone. Okay so we've now done the majority of the hard work, which is finding out our unknown radiuses and heights.
Now all we have to do is plug them all into the formula. So the volume of the frustum is equal to the volume of the large cone, so 1 third times pi times a radius of 10 times the height of 50, take away the volume of the smaller cone. So one third times pi times the radius of 4 squared times the height of 20. And if you're doing that by hand, that will simplify down to 5000 over 3 pi minus 320 over 3 pi. So our answer would be 1560 pi centimeters cubed. Or if you had a calculator, and so wanted to leave it to 3 significant figures, that would be 4900cm3.
Anyway, that's the end of this video, so I hope that's all made sense. If you want to practice questions on this stuff or anything else in science or maths, then head over to our platform by clicking the link in the top right corner of this screen, and we'll see you next time!