How to Calculate the Volume of Cones

TenMarks teaches you how to calculate the volume of cone.

In this lesson, let’s learn about volume of cones. We are given two problems. The first one wants us to look at a situation where Anne built two cones of sand on the beach. The diameter of the smaller cone is given to us and its height and the larger cone is given to us and its height. As you can see we’ve drawn out the pictures. We need to find how many cubit inches of sand that Anne used for each cone. That’s the volume. We need to express that in terms of Pi and find out how many times more or greater is the volume of the larger cone compared to the smaller cone. There is a second problem as well which we’ll come back in a few minutes and do. Let’s do the first one first. So it says Anne built two cones out of sand on the beach. The diameter of the smaller cone and the larger cone and both heights are given to us. The thing that we need to remember is volume of the cone is one-third times the area of its space times the height. Now we know that in case of a cone the base is a circle. So the area of the base is Pi times the square of the radius. The volume of the cone becomes one-third times Pi multiplied by the square of the radius times its height. That’s the formula we’re going to use. Now, let’s do Part A. How many cubic inches of sand did Anne use for each cone? So, let’s called them cone 1 and cone 2. In cone 1, what do we know? The volume becomes 1/3 times Pi times, well the radius is half of the diameter so the radius is 6 inches here and the radius is three inches here. so, in cone 1, the radius is 6 inches, so multiplying it by itself, it is six squared times the height which is eight inches and the volume will always be in cubic units. This is square, this is inches, and so square times inches is inches cube. So let’s multiply this. That’s 1/3 times Pi times 36 times eight which is 96 cubic inches. That’s all aligned and do volume two equals one-third times Pi times instead of six squared, the radius is three inches so that would be three squared times the height being four inches exactly the same thing which becomes 12 inches cube. So the volume of the first cone is 96 cubic inches. The volume of the second cone is 12 cubic inches. So we’ve done Part A. Part B says how many times greater is the volume of larger versus the smaller. So, Part B, is how many times greater? We have to find the ratio. So volume one by volume two is what we need to determine. Volume one is 96 cubic inches over volume two which is12 cubic inches, so it’s eight times larger, eight times squared. That’s what we’re looking for. Now, let’s look at the second problem. The key thing to remember in the first one is the formula of how we compute the volume of a cone. Let’s look at the second problem. It says the cone has a radius of three meters. So R equals three meters and height equals ten meters. We need to explain whether doubling the height. If we double the height, it will have the same effect on the volume as doubling the radius. So, let’s take scenario one where we double the height. We know the formula for volume is 1/3 times Pi times R squared times height. All of these are multiplications. So if I double the height, the volume will become 1/3 times Pi times the radius squared times twice the height. Let’s solve this out which becomes Pi times r squared times 2h over three. Now, let’s look at scenario two where we double the radius. In this case, the volume is 1/3 times Pi times inch of R. I’ve got two r squared times height which becomes one-third times Pi times four R squared times h which is the same as one third times Pi times r squared times four h. So, as we can see, when I compare this to this, this is twice, right? So this is four times the volume of the original one and this is twice the volume of original. Notice 2h, this is 4h. Everything else is the same. So this is one, yeah. 1/3 times Pi times r squared is the same. This is four times height, this is two times height. So, the volume here doubles. Here it’s four times so where the doubling of the height will have the same effect? No, it would actually quadruple in case we double the radius and double if we double the height.