Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Extension of AF3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = bh, C = d–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

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Vocabulary

composite figure

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

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.

Holt CA Course 1

10-6 Area of Irregular and Composite FiguresAdditional 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 =51.51 2

The area of the figure is about 51.5 yd2.

Holt CA Course 1

10-6 Area of Irregular and Composite FiguresCheck 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

The area of the figure is about 15 yd .2

Holt CA Course 1

10-6 Area of Irregular and Composite FiguresAdditional Example 2: Finding the Area of a Composite

Figure

Find the area of the composite figure. Use 3.14 as an estimate for p.

Use the formula for the area of a parallelogram.

Substitute 16 for b.Substitute 9 for h.

A = bh

A = 16 • 9

A = 144

Step 1: Separate the figure into smaller, familiar figures.

16 m

9 m

16 m

Step 2: Find the area of each smaller figure.

Area of the parallelogram:

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

The area of a semicircle is the area of a circle. t A = (r2)

1 2

1 2

Helpful Hint

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Additional Example 2 Continued

Find the area of the composite figure. Use 3.14 as an estimate for .

Substitute 3.14 for and 8 for r.

16 m

9 m

16 m

Area of the semicircle:

A = (r2)12__

The area of a semicircle

is the area of a circle.12

A ≈ (3.14 • 82)12

__

A ≈ (200.96) 12

__

Multiply.A ≈ 100.48

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Additional Example 2 Continued

Find the area of the composite figure. Use 3.14 as an estimate for .

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.

16 m

9 m

16 m

Holt CA Course 1

10-6 Area of Irregular and Composite FiguresCheck It Out! Example 2

Find the area of the composite figure.

Use the formula for the area of a rectangle.Substitute 8 for l.Substitute 9 for w.

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:

2 yd

9 yd

8 yd

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Check It Out! Example 2 Continued

Find the area of the composite figure.

Substitute 2 for b and 9 for h.

Area of the triangle:

A = bh12__

The area of a triangle

is the b • h.12

A = (2 • 9)12

__

A = (18) 12

__

Multiply.A = 9

2 yd

9 yd

8 yd

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Check It Out! Example 2 Continued

Find the area of the composite figure. Use 3.14 as an estimate for .

A = 72 + 9 = 81

The area of the figure is about 81 yd2.

Step 3: Add the area to find the total area.

Holt CA Course 1

10-6 Area of Irregular and Composite FiguresAdditional 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

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Additional Example 3 Continued

11 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.

Holt CA Course 1

10-6 Area of Irregular and Composite FiguresAdditional 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.

22 Make a Plan

5 ft

8 ft

4 ft

7 ft

t

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

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

Helpful Hint

Holt CA Course 1

10-6 Area of Irregular and Composite FiguresAdditional Example 3 Continued

Solve33Find the area of each smaller figure.

A = lw

A = 8 • 5

A = 40

Area of the rectangle:

Area of the trapezoid:

A = 24

A = h(b1 + b2)12

__

A = • 4(5 + 7)12__

A = • 4 (12)12__

Add the areas to find the total area.

A = 40 + 24 = 64

They need 64 ft2 of tile.

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Additional Example 3 Continued

Look Back44

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

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Check It Out! Example 3

The Franklins want to wallpaper the wall of their daughters loft. How much wallpaper will they need?

6 ft

23 ft

18 ft5 ft

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Check It Out! Example 3 Continued

11 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.

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

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 Make a Plan

6 ft

23 ft

18 ft5 ft

Holt CA Course 1

10-6 Area of Irregular and Composite FiguresCheck It Out! Example 3 Continued

Solve33Find the area of each smaller figure.

A = lw

A = 18 • 6

A = 108

Area of the rectangle:

Area of the triangle:

Add the areas to find the total area.

A = 108 + 27.5 = 135.5

They need 135.5 ft2 of wallpaper.

A = 27.5

A = bh12__

A = (5 • 11)12__

A = (55)12__

Holt CA Course 1

10-6 Area of Irregular and Composite Figures

Check It Out! Example 3 Continued

Look Back44

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