# Intermediate Geometry : Cubes

## Example Questions

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### Example Question #71 : Solid Geometry

A sphere with a radius of is cut out of a cube that has a side length of . What is the volume of the resulting figure?     Explanation:

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere. Plug in the given radius to find the volume. Next, recall how to find the volume of a cube: Plug in the given side length to find the volume of the cube. Finally, subtract the volume of the sphere from the volume of the cube. Make sure to round to places after the decimal.

### Example Question #72 : Solid Geometry The above figure shows a triangular pyramid inscribed inside a cube. The pyramid has volume 100. Give the volume of the cube.      Explanation:

Let be the length of one side of the cube. Then the base of the pyramid is a right triangle with legs of length . The area of a right triangle is half the product of the lengths of the legs, so the base has area The volume of a pyramid is one third the product of its height and the area of its base. The height of the pyramid is , so the volume is The volume of the cube is , so ,

or, equivalently,  That is, the volume of the cube is six times that of the pyramid, and, since the pyramid has volume 100, the volume of the cube is .

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