Question Video: Finding the Diameter of a Sphere given Its Surface Area Mathematics • Second Year of Preparatory School

Video Transcript

What is the diameter of a sphere whose surface area is 36𝜋 square centimeters?

In this question, we’ve been given the surface area of a sphere and asked to use this to determine its diameter. We recall that the general formula for finding the surface area of a sphere is four 𝜋𝑟 squared. So by equating these two pieces of information, we can form an equation that will enable us to determine firstly the radius of the sphere. We have the equation four 𝜋𝑟 squared equals 36𝜋. And we can now solve this equation. Firstly, we can cancel a factor of 𝜋 on each side. We can then divide each side of the equation by four to leave 𝑟 squared on the left-hand side and nine on the right-hand side. So we now have the equation 𝑟 squared is equal to nine.

We solve this equation by square rooting. And we’re only going to take the positive value here because 𝑟 has a physical meaning as the radius of the sphere. Nine is a square number, and its square root is three. So we found that the radius of the sphere is three centimeters. We must be careful, though, because it wasn’t the radius of the sphere that we were originally asked to find. It was the diameter. But that’s no problem because we know that the diameter of a sphere is twice its radius. So if the radius is three, the diameter will be six. We’ve solved the problem, and the diameter of the sphere whose surface area is 36𝜋 square centimeters is six centimeters.