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9 THE NATURE OF MEASUREMENT Copyright © Cengage Learning. All rights reserved. 9.2 Area Copyright © Cengage Learning. All rights reserved. Rectangles 3 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 4 Example 1 – Find an area What is the area of the shaded region? 5 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 6 Rectangles 7 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. 8 Parallelograms 9 Parallelograms A parallelogram is a quadrilateral with two pairs of parallel sides, as shown in Figure 9.9. Parallelograms Figure 9.9 10 Parallelograms The formula for the area of a parallelogram is the same as the formula for the area of a rectangle. 11 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 12 Example 3 – Solution cont’d b. b = 5 in. and h = 2 in. A = bh = 2 in. 5 in. = 10 in.2 13 Triangles 14 Triangles 15 Example 4 – Find the area of a triangle Find the area of each shaded region. 16 Example 4 – Solution a. b = 3 mm, h = 4 mm b. b = 5 km, h = 3 km 17 Trapezoids 18 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 19 Trapezoids 20 Example 5 – Find the area of a trapezoid Find the area of each shaded region. 21 Example 5 – Solution a. h = 4; b = 8; B = 15 The area is 46 in.2. 22 Example 5 – Solution cont’d b. h = 3; b = 19; B = 5 The area is 36 ft2. 23 Circles 24 Circles 25 Example 6 – Area of a circle or semicircle Find the area (to the nearest tenth unit) of each shaded region. 26 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. 27 Applications 28 Applications Sometimes we measure area using one unit of measurement and then we want to convert the result to another. 29 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. 30 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 31 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. 32 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. 33 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 34