# PSAT Math : How to find the volume of a cube

## Example Questions

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### Example Question #11 : How To Find The Volume Of A Cube

A cube is inscribed inside a sphere of radius 1 such that each of the eight vertices of the cube lie on the surface of the sphere.  What is the volume of the cube?

Possible Answers:

Correct answer:

Explanation:

To make this problem easier to solve, we can inscribe a smaller square in the cube.  In the diagram above, points  are midpoints of the edges of the inscribed cube.  Therefore point , a vertex of the smaller cube, is also the center of the sphere.  Point  lies on the circumference of the sphere, so .   is also the hypotenuse of right triangle .  Similarly,  is the hypotenuse of right triangle .  If we let , then, by the properties of a right triangle, .

Using the Pythagorean Theorem, we can now solve for :

Since the side of the inscribed cube is , the volume is .

### Example Question #12 : How To Find The Volume Of A Cube

If a cube has a surface area of , what is the difference between the volume of the cube and the surface area of the cube?

Possible Answers:

Correct answer:

Explanation:

If the surface area is , then the area of one face must be . Therefore, the length of one edge must be  This means that the volume of the cube is . We can now solve with:

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