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Preview Warm Up California Standards Lesson Presentation

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Warm Up Find the area of the following figures. 1. a triangle with a base of 12.4 m and a height of 5 m 2. a parallelogram with a base of 36 in. and a height of 15 in. 3. a square with side lengths of 2.5 yd 31 m2 540 in2 6.25 yd2

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Extension of AF3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = bh, C = pd–the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). Also covered: AF3.2, MG1.1, MG1.2 California Standards 1 2

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Vocabulary composite figure

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A composite figure is made up of simple geometric shapes, such as triangles and rectangles. You can find the area of composite and other irregular figures by separating them into non-overlapping familiar figures. The sum of the areas of these figures is the area of the entire figure. You can also estimate the area of irregular figures by using graph paper.

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**Teacher Example 1: Estimating the Area of an Irregular Figure**

Estimate the area of the figure. Each square represents one square yard. Count the number of filled or almost-filled squares: 47 squares. Count the number of squares that are about half-full: 9 squares. Add the number of filled squares plus ½ the number of half-filled squares: 47 + ( • 9) = =51.5 1 2 The area of the figure is about 51.5 yd2.

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**Estimate the area of the figure. Each square represents 1 yd2.**

Student Practice 1: Estimate the area of the figure. Each square represents 1 yd2. Count the number of filled or almost-filled squares: 11 red squares. Count the number of squares that are about half-full: 8 green squares. Add the number of filled squares plus ½ the number of half-filled squares: 11 + ( • 8) = = 15. 1 2 The area of the figure is about 15 yd . 2

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**Teacher Example 2: Finding the Area of a Composite Figure**

Find the area of the composite figure. Use 3.14 as an estimate for p. Step 1: Separate the figure into smaller, familiar figures. 16 m 9 m Step 2: Find the area of each smaller figure. 16 m Area of the parallelogram: A = bh Use the formula for the area of a parallelogram. A = 16 • 9 Substitute 16 for b. Substitute 9 for h. A = 144

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**The area of a semicircle is the area of a circle. t A = (pr2)**

1 2 Helpful Hint

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**Teacher Example 2 Continued:**

Find the area of the composite figure. Use 3.14 as an estimate for p. Area of the semicircle: 16 m 9 m The area of a semicircle is the area of a circle. A = (pr2) 1 2 __ 12 A ≈ (3.14 • 82) 1 2 __ Substitute 3.14 for p and 8 for r. A ≈ (200.96) 1 2 __ A ≈ Multiply.

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**Teacher Example 2 Continued:**

Find the area of the composite figure. Use 3.14 as an estimate for p. 16 m 9 m Step 3: Add the area to find the total area. A ≈ = The area of the figure is about m2.

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**Find the area of the composite figure.**

Student Practice 2: Find the area of the composite figure. Step 1: Separate the figure into smaller, familiar figures. 9 yd Step 2: Find the area of each smaller figure. Area of the rectangle: 2 yd 8 yd A = lw Use the formula for the area of a rectangle. A = 8 • 9 Substitute 8 for l. Substitute 9 for w. A = 72

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**Student Practice 2 Continued:**

Find the area of the composite figure. Area of the triangle: The area of a triangle is the b • h. 12 9 yd A = bh 1 2 __ A = (2 • 9) 1 2 __ 2 yd 8 yd Substitute 2 for b and 9 for h. A = (18) 1 2 __ A = 9 Multiply.

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**Student Practice 2 Continued:**

Find the area of the composite figure. Use 3.14 as an estimate for p. Step 3: Add the area to find the total area. A = = 81 The area of the figure is about 81 yd2.

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**Teacher Example 3: Problem Solving Application**

The Wrights want to tile their entry with one-square-foot tiles. How much tile will they need? 5 ft 8 ft 4 ft 7 ft

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**Understand the Problem**

Teacher Example 3 Continued: 1 Understand the Problem Rewrite the question as a statement. • Find the amount of tile needed to cover the entry floor. List the important information: • The floor of the entry is a composite figure. • The amount of tile needed is equal to the area of the floor.

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**Teacher Example 3 Continued:**

2 Make a Plan Find the area of the floor by separating the figure into familiar figures: a rectangle and a trapezoid. Then add the areas of the rectangle and trapezoid to find the total area. 5 ft 8 ft 4 ft 7 ft t

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**There are often several different ways to separate a composite figure into familiar figures.**

Helpful Hint

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**Teacher Example 3 Continued:**

Solve 3 Find the area of each smaller figure. Area of the rectangle: Area of the trapezoid: A = lw A = h(b1 + b2) 1 2 __ A = 8 • 5 A = • 4(5 + 7) 1 2 __ A = 40 A = • 4 (12) 1 2 __ Add the areas to find the total area. A = 24 A = = 64 They need 64 ft2 of tile.

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**Teacher Example 3 Continued:**

4 Look Back The area of the entry must be greater than the area of the rectangle (40 ft2), so the answer is reasonable.

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Student Practice 3: The Franklins want to wallpaper the wall of their daughters loft. How much wallpaper will they need? 6 ft 23 ft 18 ft 5 ft

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**Understand the Problem**

Student Practice 3 Continued: 1 Understand the Problem Rewrite the question as a statement. • Find the amount of wallpaper needed to cover the loft wall. List the important information: • The wall of the loft is a composite figure. • The amount of wallpaper needed is equal to the area of the wall.

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**Student Practice 3 Continued:**

2 Make a Plan Find the area of the wall by separating the figure into familiar figures: a rectangle and a triangle. Then add the areas of the rectangle and triangle to find the total area. 6 ft 23 ft 18 ft 5 ft

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**Student Practice 3 Continued:**

Solve 3 Find the area of each smaller figure. Area of the rectangle: Area of the triangle: A = lw A = bh 1 2 __ A = 18 • 6 A = (5 • 11) 1 2 __ A = 108 A = (55) 1 2 __ Add the areas to find the total area. A = 27.5 A = = 135.5 They need ft2 of wallpaper.

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**Student Practice 3 Continued:**

4 Look Back The area of the wall must be greater than the area of the rectangle (108 ft2), so the answer is reasonable.

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Lesson Quiz Find the perimeter and area of each figure. 1. 2. 6 cm cm, cm2 8 cm 10 ft 39.1 ft, 84 ft2 8.1 ft 7 ft 14 ft

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