# 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 cm² 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 𝑟, then its surface area, 𝑆, is given by four 𝜋𝑟 squared. Substituting 𝑆 equals 16𝜋, we have 16𝜋 equal to four 𝜋𝑟 squared. Now we will solve for 𝑟 by dividing through by four 𝜋. This gives us four equal to 𝑟 squared. Then, we can obtain 𝑟 by taking the square root of four. So, 𝑟 equals two centimeters.

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