EXAMPLE 3
12 in.
1 foot .
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Chapter 9 An Introduction to Geometry |
Since a rectangle has two lengths l and two widths w, its perimeter P is given by l w l w, or 2l 2w.
Perimeter of a Rectangle
If a rectangle has length l and width w, its perimeter P is
given by the formula
w
P 2l 2w
l
Self Check 2
Find the perimeter of the triangle shown below, in inches.
2 ft
Now Try Problem 21
Caution! When finding the perimeter of a polygon, the lengths of the sides must be expressed in the same units.
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EXAMPLE 2 |
Find the perimeter of the rectangle shown on |
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the right, in inches. |
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Strategy We will express the width of the rectangle in inches and |
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3 ft |
then use the formula P 2l 2w to find the perimeter of the |
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figure. |
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8 in. |
WHY We can only add quantities that are measured in the same |
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units. |
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Solution Since 1 foot 12 inches, we can convert 3 feet to inches by multiplying 3 feet by the unit conversion factor
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3 ft |
3 ft |
12 in. |
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Multiply by 1: 12 in. 1. |
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1 ft |
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1 ft |
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3 ft |
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12 in. |
Write 3 ft as a fraction. Remove the common units of feet from |
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1 ft |
the numerator and denominator. The units of inches remain. |
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36 in. |
Do the multiplication. |
The width of the rectangle is 36 inches. We can now substitute 8 for l , the length, and 36 for w, the width, in the formula for the perimeter of a rectangle.
P 2l 2w |
This is the formula for the perimeter of a rectangle. |
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36 |
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P 2(8) 2(36) |
Substitute 8 for l, the length, and 36 for w, the width. |
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72 |
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16 72 |
Do the multiplication. |
16 |
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88 |
Do the addition. |
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72 |
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The perimeter of the rectangle is 88 inches. |
88 |
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Self Check 3
The perimeter of an isosceles triangle is 58 meters. If one of its sides of equal length is 15 meters long, how long is its base?
Now Try Problem 25
Structural Engineering The truss shown below is made up of three parts that form an isosceles triangle. If 76 linear feet of lumber were used to make the truss, how long is the base of the truss?
20 ft
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9.7 Perimeters and Areas of Polygons |
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Analyze |
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• The truss is in the shape of an isosceles triangle. |
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• One of the sides of equal length is 20 feet long. |
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• The perimeter of the truss is 76 feet. |
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• What is the length of the base of the truss? |
Find |
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Form an Equation We can let b equal the length of the |
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20 |
base of the truss (in feet).At this stage, it is helpful to draw
a sketch. (See the figure on the right.) If one of the sides of |
b |
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equal length is 20 feet long, so is the other. |
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Because 76 linear feet of lumber were used to make the triangular-shaped |
truss, |
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The length |
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the length |
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the length |
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of the base |
plus |
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of the |
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of one side |
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of the truss |
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other side |
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b |
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20 |
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20 |
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Solve
b 20 20 76
b 40 76 Combine like terms.
b 36 To isolate b, subtract 40 from both sides.
State The length of the base of the truss is 36 ft.
Check If we add the lengths of the parts of the truss, we get 36 ft 20 ft 20 ft 76 ft. The result checks.
the perimeter of the truss.
76
7640 36
Using Your CALCULATOR Perimeters of Figures That Are Combinations of Polygons
To find the perimeter of the figure shown below, we need to know the values of x and y. Since the figure is a combination of two rectangles, we can use a calculator to see that
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20.25 cm |
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y cm |
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x cm |
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12.5 cm |
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4.75 cm |
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10.17 cm |
x 20.25 10.17 |
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y 12.5 4.75 |
10.08 cm |
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7.75 cm |
The perimeter P of the figure is
P 20.25 12.5 10.17 4.75 x y
P 20.25 12.5 10.17 4.75 10.08 7.75
We can use a scientific calculator to make this calculation.
20.25 |
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12.5 |
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10.17 |
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4.75 |
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10.08 |
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7.75 |
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65.5 |
The perimeter is 65.5 centimeters.
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Chapter 9 An Introduction to Geometry |
2 Find the area of a polygon.
The area of a polygon is the measure of the amount of surface it encloses. Area is measured in square units, such as square inches or square centimeters, as shown below.
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1 in. |
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1 cm |
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1 in. |
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1 in. |
1 cm |
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1 cm |
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1 cm |
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1 in. |
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One square inch |
One square centimeter |
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(1 in.2) |
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(1 cm2) |
In everyday life, we often use areas. For example,
•To carpet a room, we buy square yards.
•A can of paint will cover a certain number of square feet.
•To measure vast amounts of land, we often use square miles.
•We buy house roofing by the “square.” One square is 100 square feet.
The rectangle shown below has a length of 10 centimeters and a width of 3 centimeters. If we divide the rectangular region into square regions as shown in the figure, each square has an area of 1 square centimeter—a surface enclosed by a square measuring 1 centimeter on each side. Because there are 3 rows with 10 squares in each row, there are 30 squares. Since the rectangle encloses a surface area of 30 squares, its area is 30 square centimeters, which can be written as 30 cm2.
This example illustrates that to find the area of a rectangle, we multiply its length by its width.
10 cm
3 cm
1 cm2
Caution! Do not confuse the concepts of perimeter and area. Perimeter is the distance around a polygon. It is measured in linear units, such as centimeters, feet, or miles. Area is a measure of the surface enclosed within a polygon. It is measured in square units, such as square centimeters, square feet, or square miles.
In practice, we do not find areas of polygons by counting squares. Instead, we use formulas to find areas of geometric figures.
Self Check 1
Self Check 4
Self Check 5