Video: Finding the Volume of a Sphere given the Circumference of Its Great Circle

Video Transcript

Find, to the nearest tenth, the volume of a sphere given that the circumference of its great circle is 90𝜋 inches.

So first of all, what do we mean by the great circle of a sphere? Well, formally, it is the intersection of the sphere and any plane — that’s a two-dimensional slice — which passes through the center of the sphere. On our diagram, this is one such great circle. But in fact, there are infinitely many depending on the angle of the plane we draw.

Now we know that, in order to find the volume of any sphere, we use the formula four-thirds 𝜋𝑟 cubed. So in order to answer this question, we need to calculate the radius of our sphere. We can do this using the information given about the circumference of the great circle because we know that the circumference of any circle is found using the formula 𝜋𝑑 or two 𝜋𝑟.

We can therefore form an equation using the version of the circumference formula that involves 𝑟. Two 𝜋𝑟 is equal to 90𝜋. And we can solve this equation in order to determine the radius of the sphere. First, we cancel a factor of 𝜋 from each side, giving two 𝑟 equals 90. And then we divide by two to find that 𝑟 is equal to 45. The units for this will be inches. Finally, we can substitute this value for the radius into our formula for the volume of a sphere, giving four-thirds multiplied by 𝜋 multiplied by 45 cubed.

As we’re asked to give our answer to the nearest tenth, it’s reasonable to assume we can use a calculator to help with this question. So evaluating on a calculator gives 121,500𝜋. Or as a decimal, this is equivalent to 381,703.5074. As we’re rounding to the nearest tenth, our deciding digit is in the hundredths column. That’s a zero. So we’re rounding down.

So we find that the volume of the sphere whose great circle has a circumference of 90𝜋 inches, to the nearest tenth, is 381,703.5 cubic inches.