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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:

V = l • w • h
S = 2wl + 2hl + 2wh

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:

V = x3
S = 6x2

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

(A) 6
(B) 10
(C) 18
(D) 36
(E) 48

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:

V = πr2h
S = 2πrh + 2πr2