The volume of a cone is 441𝜋 cubic inches and its height is 12 inches. Find its diameter.
So we’ve been given two key pieces of information about this cone: its volume and its height. And our objective is to find the diameter of the cone. Let’s recall, first of all, the formula for calculating the volume of a cone. It is one-third 𝜋𝑟 squared ℎ, where 𝑟 is the radius of the cone and ℎ is the height.
Remember, we know both the volume and the height of this cone. So we can form an equation, which we can then solve, in order to calculate the radius. Once we know the radius, we just need to recall that the diameter is twice the radius. So whatever value we get for 𝑟, we need to double it to find the value of 𝑑.
Let substitute in the values we know. So we have 441𝜋, for the volume, is equal to one-third 𝜋𝑟 squared multiplied by 12. Now there’s a factor of 𝜋 on both sides of this equation. So these can cancel each other out directly.
On the right-hand side, there’s a three in the denominator and a 12 in the numerator. So these will simplify to just give an overall factor of four. Therefore, the equation simplifies to 441 is equal to four 𝑟 squared. Now there are two ways that you could proceed from here.
Remember, ultimately we’re looking to calculate the value of 𝑑 which is two 𝑟. And two 𝑟 is what I would get if I were to square root the right-hand side of the equation. I also notice that 441 is a square number. It’s 21 squared. And therefore, in this case, square rooting both sides of the equation seems like a sensible thing to do.
Remember you must square root each factor on the right-hand side, not just the 𝑟 squared. When we square root 441, we get 21. When we square root four we get two. And when we square root 𝑟 squared, we get 𝑟. So we have 21 is equal to two 𝑟. Remember two 𝑟 is equivalent to 𝑑, the diameter. So we have the diameter of the cone is 21 inches.
If you didn’t spot that 441 was a square number, then there would have been another way of proceeding at this stage here. We could have divided both sides of the equation by four. This would’ve given 𝑟 squared is equal to 441 over four which as a decimal is 110.25. Next, we could have taken the square root of this value. And you would certainly have needed your calculator in order to work this out.
And it would tell you that 𝑟 is equal to 10.5. Once you know the radius, remember you need to double it in order to get the diameter. And doubling 10.5 gives the value of 21 that we’ve already calculated. So spotting 441 was a square number is perhaps the smarter approach.
But if you didn’t spot that and if you have access to a calculator, then the other method would of course bring you to the same point. Our answer to the problem then is the diameter of the cone is 21 inches.