# Volume

The volume of a rectangular solid (a box) is the product of the length, width, and height. The surface area is the sum of the area of the six faces:

If the length, width, and height of a rectangular solid (a box) are the same, it is a cube. Its volume is the cube of one of its sides, and its surface area is the sum of the areas of the six faces:

Example

The volume of the cube is x and its surface area is x. What is the length of an edge of the cube?

1. 6
2. 10
3. 18
4. 36
5. 48
Solution

Let e be the length of an edge of the cube.

Recall that the volume of a cube is e3 and its surface area is 6e2.

Since we are given that both the volume and the surface area are x, these expressions are equal:

e3 = 6e2
e3 – 6e2 = 0
e2(e – 6) = 0
e2 = 0 or  e – 6 = 0
e = 0 or  e = 6

We reject e = 0 since in that case no cube would exist.

Hence, e = 6 and the answer is (A).

The volume of a cylinder is V = πr2h, and the lateral surface (excluding the top and bottom) is S = 2πrh, where r is the radius and h is the height: