PSAT Math : How to find the volume of a cube

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #32 : Solid Geometry

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:

Cube

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 #221 : Psat Mathematics

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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