Question Video: Finding the Radius of a Sphere given Its Volume Mathematics • Second Year of Preparatory School

Video Transcript

Find the radius of a sphere whose volume is nine over two 𝜋 cubic centimeters.

In this problem, we’ve been given the volume of a sphere and asked to work backwards to determine its radius. Let’s begin by recalling the formula we use for calculating the volume of a sphere. It’s this, four-thirds 𝜋𝑟 cubed, where 𝑟 represents the radius of the sphere.

Now as we’ve been given the volume and we know the general formula for working it out, we can set these two values or expressions equal to one another to give an equation. We have four-thirds 𝜋𝑟 cubed equals nine over two 𝜋. And in order to determine the radius of the sphere, we simply need to solve this equation.

First, we notice that there’s a factor of 𝜋 on each side of the equation. So we can cancel this. Or we can think of this as dividing through by 𝜋, to give four-thirds 𝑟 cubed equals nine over two. Next, we need to divide each side of the equation by four-thirds in order to leave 𝑟 cubed on its own on the left-hand side.

We recall that dividing by a fraction is equivalent to multiplying by the reciprocal of that fraction. So to divide by four-thirds, we can multiply each side of the equation by three-quarters. Doing so will eliminate the factor of four-thirds on the left-hand side, leaving just 𝑟 cubed. And on the right-hand side, we have nine over two multiplied by three over four. We multiply the numerators, giving 27, and multiply the denominators, giving eight. So we find that 𝑟 cubed is equal to 27 over eight. To find the value of 𝑟, we need to perform the inverse or opposite of cubing, which is cube rooting. So we find that 𝑟 is equal to the cubed root of 27 over eight.

Now at this point, we remember that, in order to find the cubed root of a fraction, we can find the cubed root of the numerator over the cubed root of the denominator. So we have that 𝑟 equals the cubed root of 27 over the cubed root of eight. And these are both integer values. The cubed root of 27 is three, and the cubed root of eight is two. So we find that the radius of this sphere is three over two or 1.5. And as the units for the volume were cubic centimeters, the units for the radius will be centimeters.

Now of course, we weren’t asked for it in this problem. But if we needed to calculate the diameter of the sphere, we just need to recall that the diameter is twice the radius. So if the radius is three over two centimeters, then the diameter is twice this. The diameter of the sphere is three centimeters. We’ve completed the problem there. The radius of the sphere whose volume is nine over two 𝜋 cubic centimeters is three over two centimeters.