**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* = 2*wl* + 2*hl *+ 2*wh*

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* = *x*^{3}

S = 6*x*^{2}

**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

^{3}and its surface area is 6e

^{2}. Since we are given that both the volume and the surface area are x, these expressions are equal:

*e*

^{3}= 6

*e*

^{2}

*e*

^{3}-6

*e*

^{2}= 0

*e*

^{2}(

*e*-6) = 0

*e*

^{2}= 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 *= *πr*^{2}*h* , and the lateral surface (excluding the top and bottom) is *S* = 2*πrh*, where r is the radius and h is the height:

*V* = *πr*^{2}*h*

*S* = 2*πrh* + 2*πr*^{2}