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Question Video: Relating the Volumes of a Cube and a Sphere Mathematics • 8th Grade

If a sphere is inscribed in a cube of volume 8 cmΒ³, what is the volume of the sphere?

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Video Transcript

If a sphere is inscribed in a cube of volume eight cubic centimeters, what is the volume of the sphere?

The key to this question is being able to relate the dimensions of the cube to the sphere. Since the sphere is inscribed in the cube, this means that the sphere is touching each face of the cube without any gaps. As such, the diameter 𝑑 of the sphere is equal to the length 𝑙 of the cube. Alternatively, the radius π‘Ÿ is half of the length of the cube.

Next, we recall that the volume of any cube is equal to its side length cubed. And in this question, we’re told that this volume is eight cubic centimeters. This means that 𝑙 cubed is equal to eight. We can then cube root both sides such that 𝑙 is equal to two. The side length of the cube is therefore equal to two centimeters.

We have already established that the radius of the sphere is half of this. And this is therefore equal to one centimeter. We can calculate the volume of any sphere when we know its radius. The volume is equal to four-thirds πœ‹π‘Ÿ cubed. If we let the volume of our sphere be 𝑉, we have 𝑉 is equal to four-thirds πœ‹ multiplied by one cubed. And this is simply equal to four-thirds πœ‹. We can therefore conclude that if the volume of the cube is eight cubic centimeters and the sphere is inscribed in the cube, then its volume is four-thirds πœ‹ cubic centimeters.

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