California Standards

1. California Standards 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 1 2

2. Vocabulary composite figure

3. 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.

4. The area of the figure is about 51.5 yd2. Additional 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) = 47 + 4.5 =51.5 1 2

5. 2 The area of the figure is about 15 yd . Check It Out! Example 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) = 11 + 4 = 15. 1 2

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

7. Helpful Hint The area of a semicircle is the area of a circle. tA = (pr2) 1 2 1 2

8. 16 m 1 1 1 __ __ __ 2 2 2 9 m 16 m A ≈ 100.48 A = (pr2) Area of the semicircle: A ≈ (3.14 • 82) A ≈ (200.96) Additional Example 2 Continued Find the area of the composite figure. Use 3.14 as an estimate for p. The area of a semicircle is the area of a circle. 12 Substitute 3.14 for p and 8 for r. Multiply.

9. 16 m 9 m 16 m A ≈ 144 + 100.48 = 244.48 The area of the figure is about 244.48 m2. Step 3: Add the area to find the total area. Additional Example 2 Continued Find the area of the composite figure. Use 3.14 as an estimate for p.

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

11. 1 1 1 __ __ __ 2 2 2 The area of a triangle is the b•h. 12 A = 9 Area of the triangle: A = bh A = (2•9) A = (18) Check It Out! Example 2 Continued Find the area of the composite figure. 9 yd 2 yd 8 yd Substitute 2 for b and 9 for h. Multiply.

12. A = 72 + 9 = 81 The area of the figure is about 81 yd2. Step 3: Add the area to find the total area. Check It Out! Example 2 Continued Find the area of the composite figure. Use 3.14 as an estimate for p.

13. 5 ft 8 ft 4 ft 7 ft Additional Example 3: Problem Solving Application The Wrights want to tile their entry with one-square-foot tiles. How much tile will they need?

14. 1 Understand the Problem Additional Example 3 Continued 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.

15. Make a Plan 5 ft 8 ft t 4 ft 2 7 ft Additional Example 3 Continued 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.

16. Helpful Hint There are often several different ways to separate a composite figure into familiar figures.

17. 3 Solve 1 1 1 __ __ __ 2 2 2 Area of the rectangle: A = 40 + 24 = 64 A =lw A = 8 •5 A = 40 Area of the trapezoid: A = • 4(12) A = h(b1 + b2) They need 64 ft2 of tile. A = • 4(5 + 7) Add the areas to find the total area. A = 24 Additional Example 3 Continued Find the area of each smaller figure.

18. 4 Additional Example 3 Continued Look Back The area of the entry must be greater than the area of the rectangle (40 ft2), so the answer is reasonable.

19. 5 ft 18 ft 6 ft 23 ft Check It Out! Example 3 The Franklins want to wallpaper the wall of their daughters loft. How much wallpaper will they need?

20. 1 Understand the Problem Check It Out! Example 3 Continued 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.

21. Make a Plan 5 ft 18 ft 6 ft 23 ft 2 Check It Out! Example 3 Continued 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.

22. 3 Solve 1 1 1 __ __ __ 2 2 2 Area of the rectangle: A = bh A =lw A = 18 •6 A = 108 Area of the triangle: A = (5 •11) A = 108 + 27.5 = 135.5 They need 135.5 ft2 of wallpaper. A = 27.5 A = (55) Add the areas to find the total area. Check It Out! Example 3 Continued Find the area of each smaller figure.

23. 4 Check It Out! Example 3 Continued Look Back The area of the wall must be greater than the area of the rectangle (108 ft2), so the answer is reasonable.