# Question Video: Relating the Volume of the Cube and the Surface Area of an Inscribed Sphere Mathematics • 8th Grade

A sphere of surface area 16π cmΒ² is inscribed in a cube. Find the volume of the cube.

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### Video Transcript

A sphere of surface area 16π square centimeters is inscribed in a cube. Find the volume of the cube.

The key to this question is being able to relate the dimensions of the cube and the sphere. We note that the sphere is inscribed in the cube, which means that the sphere is touching each face of the cube without any gaps. Therefore, the diameter of the sphere must equal the length of the cube. In other words, the length of the cube is two times the radius. This is shown in the figure.

We have been given the surface area of the sphere, 16π square centimeters. Recall that if the radius of a sphere is π, then its surface area, π, is given by four ππ squared. Substituting π equals 16π, we have 16π equal to four ππ squared. Now we will solve for π by dividing through by four π. This gives us four equal to π squared. Then, we can obtain π by taking the square root of four. So, π equals two centimeters.

We are looking for the volume of the cube, which is given by the formula π equals π cubed. As mentioned earlier, the length of the cube is two times the radius of the inscribed sphere, so we have π equal to four centimeters. Now that we have the length of the cube, we can obtain its volume by substituting π equals four. So, the volume of the cube described is 64 cubic centimeters.