**Example 1**

What is the surface area of a cube with an edge that measures 7 in.?

*Read and understand the question*. This question is looking for the surface area of a cube when the measure of the edge of the cube is given.

*Make a plan*. Use the formula *SA* = 6*e*^{2}and substitute the value of *e*. Remember that the surface area will be represented in square units.

*Carry out the plan*. Substitute into the formula to get *SA* = 6(7)^{2}. Evaluate the exponent first to get *SA* = 6(49). Multiply to find the surface area: *SA* = 294 in.^{2}.

*Check your answer*. To check your answer, divide the total surface area by 6: 294 ÷ 6 = 49. Then, take the positive square root of 49 to find the measure of the edge of the cube. The positive square root of 49 is 7, so this result is checking.

**Example 2**

What is the measure of the edge of a cube with a total surface area of 54 ft.^{2}?

*Read and understand the question*. This question is looking for the measure of the edge of a cube when the total surface area is known.

*Make a plan*. Use the formula for the surface area of a cube, and work backward to solve for *e*.

*Carry out the plan*. Substitute the values into the formula to get 54 = 6*e*^{2}. Divide each side of the equation by 6 to get 9 = *e*^{2}. Take the positive square root of each side of the equation to find the value of *e*: *e* = 3 ft.

*Check your answer*. To check this answer, substitute the value of e into the surface area formula. The formula becomes *SA* = 6(3)^{2}. Evaluate the exponent to get *SA* = 6(9). Multiply to get a surface area of 54 ft.^{2}. This result is checking.

**Solids With Curved Surfaces**

**Cylinders**

A cylinder is a solid with two circles as the bases. A right cylinder has the height perpendicular to the bases; in other words, the cylinder is upright and not tilted at all.

To find the surface area of a cylinder, imagine a label on a soup can. If the label is peeled off and flattened, it is in the shape of a rectangle. The height of this rectangle is the height of the cylinder, and the base of this rectangle is the circumference of the base of the cylinder. (See the following figure.) The area of this rectangle can be expressed as π*dh*. The area of each of the circular bases can be found by the expression π*r*^{2}, the formula for the area of a circle.

Thus, the formula for the surface area of a cylinder is *SA* = 2π*r*^{2}+ π*dh*, where *r* is the radius of the base, *d* is the diameter of the base, and *h* is the height of the cylinder.

ExampleWhat is the surface area of a cylinder with a height of 8 m and a base with a radius of 6 m? (Leave your answer in terms of π.)

*Read and understand the question*. This question asks for the surface area of a cylinder when the dimensions are given.

*Make a plan*. Use the formula for the surface area of a cylinder, and substitute the known values. Evaluate to find the surface area.

*Carry out the plan*. Because the radius is 6 m, the diameter is 2 × 6, or 12 m. The formula *SA* = 2*πr*^{2}+ *πdh* becomes *SA* = 2*π*(6)^{2}+ *π*(12)(8). Evaluate the exponents and multiply to get *SA* = 72*π* + 96*π*. Combine like terms to get *SA* = 168*π*m^{2}.

*Check your answer*. Substitute again into the formula to double-check your solution. The formula is

*SA*= 2π(6)

^{2}+ π(12)(8) = 2π(36) + 96π = 72π + 96π = 168π

This answer is checking.

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