Transcript Find an area

9
THE NATURE
OF MEASUREMENT
Copyright © Cengage Learning. All rights reserved.
9.2
Area
Copyright © Cengage Learning. All rights reserved.
Rectangles
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Rectangles
A square yard is a measure of area. To measure the area
of a plane figure, you fill it with square units.
(See Figure 9.7.)
a. 1 square inch (actual size);
abbreviated 1 sq in. or 1 in.2
b. 1 square centimeter (actual
size); abbreviated 1 sq cm or
1 cm2
Common units of measurement for area
Figure 9.7
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Example 1 – Find an area
What is the area of the shaded region?
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Example 1 – Solution
To find the area, count the number of square units in each
region.
a. You can count the number of square centimeters in the
shaded region; there are 315 squares. Also notice:
Across Down
21 cm  15 cm = 21  15 cm  cm = 315 cm2
b. The shaded region is a square foot. You can count 144
square inches inside the region. Also notice:
Across
Down
12 in.  12 in. = 144 in.2
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Rectangles
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Example 2 – Area of a square yard in square feet
How many square feet (see Figure 9.8) are there in a
square yard?
1 yd2 = 9 ft2
Figure 9.8
Solution:
Since 1 yd = 3 ft, we see from Figure 9.8 that
1 yd2 = (3 ft)2
= 9 ft2
Substitute.
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Parallelograms
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Parallelograms
A parallelogram is a quadrilateral with two pairs of parallel
sides, as shown in Figure 9.9.
Parallelograms
Figure 9.9
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Parallelograms
The formula for the area of a parallelogram is the same as
the formula for the area of a rectangle.
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Example 3 – Find the area of a parallelogram
Find the area of each shaded region.
Solution:
a. b = 3 m and h = 6 m
A = bh
= 3m  6m
= 18m2
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Example 3 – Solution
cont’d
b. b = 5 in. and h = 2 in.
A = bh
= 2 in.  5 in.
= 10 in.2
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Triangles
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Triangles
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Example 4 – Find the area of a triangle
Find the area of each shaded region.
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Example 4 – Solution
a. b = 3 mm, h = 4 mm
b. b = 5 km, h = 3 km
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Trapezoids
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Trapezoids
A trapezoid is a quadrilateral with two sides parallel. These
sides are called the bases, and the perpendicular distance
between the bases is the height. We can find the area of a
trapezoid by finding the sum of the areas of two triangles,
as shown in Figure 9.10.
Trapezoid
Figure 9.10
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Trapezoids
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Example 5 – Find the area of a trapezoid
Find the area of each shaded region.
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Example 5 – Solution
a. h = 4; b = 8; B = 15
The area is 46 in.2.
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Example 5 – Solution
cont’d
b. h = 3; b = 19; B = 5
The area is 36 ft2.
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Circles
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Circles
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Example 6 – Area of a circle or semicircle
Find the area (to the nearest tenth unit) of each shaded
region.
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Example 6 – Solution
a. Using a calculator: 92  254.4690049. To the nearest
tenth, the area is 254.5 yd2.
b. Notice that the shaded portion is only half the area of
circle.Using a calculator: 52 /2  39.26990817. To the
nearest tenth, the area is 39.3 m2.
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Applications
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Applications
Sometimes we measure area using one unit of
measurement and then we want to convert the result to
another.
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Example 7 – Carpet purchase
Suppose your living room is 12 ft by 15 ft and you want to
know how many square yards of carpet you need to cover
this area.
Solution:
Method I.
A = 12 ft  15 ft
= 180 ft2
= 180  1 ft2
= 20 yd2
Since 1 yd = 3 ft
1 yd2 = (1 yd)  (1 yd)
1 yd2 = (3 ft)  (3 ft)
1 yd2 =9 ft2
yd2 = 1 ft2 Divide both
sides by 9.
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Example 7 – Solution
Method II.
cont’d
Change feet to yards to begin the problem:
12 ft = 4 yd
and
15 ft = 5 yd
A = 4 yd  5 yd
= 20 yd2
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Applications
If the area is large, as with property, a larger unit is needed.
This unit is called an acre.
This definition leads us to a procedure for changing square
feet to acres.
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Example 8 – Find the number of acres
How many acres are there in a rectangular property
measuring 363 ft by 180 ft?
Solution:
Begin with an estimate: Estimate: 363  400 and
180  200, so area is about 400  200 = 80,000.
Thus, the number in acres is under (since our estimate
numbers are over) 80,000  40,000 = 2 acres.
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Example 8 – Solution
cont’d
Now, we carry out the actual computation:
A = 363  180 ft
= 65,340 ft2
= 65,340  43,560 acres
= 1.5 acres
By calculator
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