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# Surface Area Word Problems Study Guide

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Updated on Oct 3, 2011

## Introduction to Surface Area Word Problems

Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.

—BERTRAND RUSSELL (1872–1970)

This chapter will include details on common geometric solids and the use of the formulas needed to find the surface area of each figure. Strategies on solving word problems with surface area will also be modeled and explained.

### Geometric Solids

Geometric solids are three-dimensional figures that are defined by the number of faces, edges, and vertices that they have.

#### Tip:

Here are some important terms to know about geometric solids.

Face: the two-dimensional "sides" of a figure
Edge: the line segment formed where two faces meet
Vertices: the points where two or more edges meet

The surface area of a three-dimensional figure is the number of square units it takes to cover each face of the figure. You can think of it as the outer cover of the object.

## Rectangular Prisms

A rectangular prism is a geometric solid that has a rectangle for each of the faces. There are three dimensions in this solid: a length, a width, and a height. This is shown in the following figure.

In a rectangular prism, each of the opposite faces is congruent. To find the surface area, find the area of three of the rectangular faces and multiply each by two to represent the side opposite. Then, add these values together. The formula is SA = 2lw + 2lh + 2wh where w is the width, l is the length, and h is the height of the prism.

#### Example

What is the volume of a rectangular prism with a length of 5 m, a width of 6 m, and a height of 10 m?

Read and understand the question. This question is looking for the surface area of a rectangular prism. Each of the three dimensions is known.

Make a plan. Use the formula SA = 2lw + 2lh + 2wh, and substitute the given values for the length, width, and height.

Carry out the plan. The formula becomes SA = 2(5)(6) + 2(5)(10) + 2(6)(10). Multiply to get SA = 60 + 100 + 120. Add to get a surface area of 280 m2.

Check your answer. Substitute the values into the formula again to doublecheck your solution. The formula SA = 2(5)(6) + 2(5)(10) + 2(6)(10) simplifies to 2(30) + 2(50) + 2(60) = 60 + 100 + 120 = 280. This answer is checking.

#### Tip:

Surface area is a measure of area, so it is always calculated in square units.

## A Special Prism: Cubes

A cube is a special type of rectangular prism. The faces of a cube are six congruent squares, as shown in the following figure.

The area of each face can be found by multiplying the length of each edge of the cube by itself. To find the surface area of a cube, find the area of one face and multiply this area by six to represent all of the faces. Therefore, the formula is SA = 6e2, where e is the length of an edge of the cube.

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