Question

Consider a sphere of radius \(r\). What is the length of a side of a cube that has the same surface area as the sphere?

Short answer

2. What is the formula for the surface area of a cube? 3. If the surface area of a sphere and cube are equal, what is the equation relating their side length and radius? 4. How is the length of a side of the cube found once the equation is set up?

Step-by-step solution

Find the surface area of the sphere

We are given the radius of the sphere (\(r\)) and the formula for the surface area of a sphere is \(4\pi r^2\). Therefore, the surface area of the sphere is: Sphere_Surface_Area \(= 4\pi r^2\)

Find the surface area of the cube

Let \(s\) be the length of a side of the cube. The formula for the surface area of a cube is \(6s^2\). So the surface area of the cube is: Cube_Surface_Area \(= 6s^2\)

Set the surface area of the sphere equal to the surface area of the cube

We need to find the length of a side of a cube (\(s\)) that has the same surface area as the sphere. So we will set the two surface areas equal to each other: \(4\pi r^2 = 6s^2\)

Solve for \(s\)

Divide both sides of the equation by 6 to isolate the variable \(s^2\): \(s^2 = \dfrac{4\pi r^2}{6}\) Now take the square root of both sides to find the value of \(s\): \(s = \sqrt{\dfrac{4\pi r^2}{6}}\) So the length of a side of a cube that has the same surface area as the sphere is: \(s = \sqrt{\dfrac{4\pi r^2}{6}}\)