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Lesson 3.1.4Lesson 3.1.4

Areas of

Complex Shapes

Areas of

Complex Shapes

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Lesson

3.1.4Areas of Complex ShapesAreas of Complex Shapes

California Standard:Measurement and Geometry 2.2Estimate and compute the area of more complex or irregular two- and three- dimensional figures by breaking the figures down into more basic geometric objects.

What it means for you:You’ll use the area formulas for regular shapes to find the areas of more complex shapes.

Key words:• complex shape• addition• subtraction

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Areas of Complex ShapesAreas of Complex ShapesLesson

3.1.4

You’ve practiced finding the areas of regular shapes.

Now you’re going to use what you’ve learned to find areas of more complex shapes.

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Areas of Complex ShapesAreas of Complex Shapes

Complex Shapes Can Be Broken into Parts

Lesson

3.1.4

There are no easy formulas for finding the areas of complex shapes. However, complex shapes are often made up from simpler shapes that you know how to find the area of.

To find the area of a complex shape you:

1) Break it up into shapes that you know how to find the area of.

2) Find the area of each part separately.

3) Add the areas of each part together to get the total area.

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Areas of Complex ShapesAreas of Complex Shapes

Shapes can often be broken up in different ways. Whichever way you choose, you’ll get the same total area.

Lesson

3.1.4

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B

A

Areas of Complex ShapesAreas of Complex Shapes

Example 1

Solution follows…

Lesson

3.1.4

Find the area of this shape.

Solution

Split the shape into a rectangle and a triangle.

Area A is a rectangle.Area A = bh = 5 cm × 2 cm = 10 cm2.

Total area = area A + area B = 10 cm2 + 1.8 cm2 = 11.8 cm2

5 cm

2 cm4 cm

3.2 cm

1.8 cm

Area B is a triangle.

Area B = bh = × 2 cm × 1.8 cm = 1.8 cm2.1

2

1

2

7

50 ft

30 ft

12 ft12 ft18 ft

12 ft

Areas of Complex ShapesAreas of Complex Shapes

Guided Practice

Solution follows…

Lesson

3.1.4

1. Find the area of the complex shape shown.

32 ft

18 ft

900 + 384 + 72 = 1356 ft2

50 • 18 = 900 ft2

32 • 12 = 384 ft2

12 • 6 = 72 ft2

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Areas of Complex ShapesAreas of Complex Shapes

You Can Find Areas By Subtraction Too

Lesson

3.1.4

So far we’ve looked at complex shapes where you add together the areas of the different parts.

For some shapes, it’s easiest to find the area of a larger shape and subtract the area of a smaller shape.

= –

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Areas of Complex ShapesAreas of Complex Shapes

Example 2

Solution follows…

Lesson

3.1.4

Find the shaded area of this shape.2 cm

5 cm

20 cm

10 cm

Solution

First calculate the area of rectangle A, then subtract the area of rectangle B.

A

B

Area A = lw = 20 × 10 = 200 cm2

Area B = lw = 5 × 2 = 10 cm2

Total area = area A – area B = 200 cm2 – 10 cm2 = 190 cm2

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Areas of Complex ShapesAreas of Complex ShapesLesson

3.1.4

Since there are many stages to these questions, always explain what you’re doing and set your work out clearly.

Most problems can be solved by either addition or subtraction of areas. Use whichever one looks simpler.

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Areas of Complex ShapesAreas of Complex Shapes

Example 3

Solution follows…

Lesson

3.1.4

Find the shaded area of this shape.

Solution

First calculate the area of triangle A, then subtract the area of triangle B.

Total area = area A – area B = 621 ft2 – 42 ft2 = 579 ft2

Area B = × 12 × 7 = 42 ft21

2Area A = × 54 × 23 = 621 ft2

1

2

54 ft

23 ft 12 ft

7 ft

AB

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Areas of Complex ShapesAreas of Complex Shapes

Guided Practice

Solution follows…

Lesson

3.1.4

Use subtraction to find the areas of the shapes in Exercises 2–3.

2. 3.

0.7 m

1 m

3 m

3 m

(3 • 3.7) – (0.7 • 1) = 10.4 m2 (5 • 7) – (0.5 • 5 • 4) = 25 ft2

5 ft

7 ft

5 ft

4 ft

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Areas of Complex ShapesAreas of Complex Shapes

Guided Practice

Solution follows…

Lesson

3.1.4

Use subtraction to find the areas of the shape in Exercise 4.

4.

(100 • 30) – (30 • 20) = 2400 in2

100 in

Use subtraction to find the areas of the shape in Exercise 4.

4.100 in

30 in 20 in30 in

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Areas of Complex ShapesAreas of Complex Shapes

Independent Practice

Solution follows…

Lesson

3.1.4

Use either addition or subtraction to find the areas of the following shapes.

1. 2.

8 cm 5 cm

3 cm

2 cm

12 mm

11 mm

6 mm

7 mm

27.5 cm2 18 mm2

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Areas of Complex ShapesAreas of Complex Shapes

Independent Practice

Solution follows…

Lesson

3.1.4

Use either addition or subtraction to find the areas of the following shapes.

3. 4.

580 in2

40 in

20 in

9 in

2875 ft2

100 ft

125 ft

130 ft

50 ft 23 ft

25 ft

2875 ft2

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Areas of Complex ShapesAreas of Complex Shapes

Independent Practice

Solution follows…

Lesson

3.1.4

5. Damion needs his window frame replacing. If the outside edge of the frame is a rectangle measuring 3 ft × 5 ft and the pane of glass inside is a rectangle measuring 2.6 ft by 4.5 ft, what is the total area of the frame that Damion needs?

6. Aisha has a decking area in her backyard. Find its area, if the deck is made from six isosceles triangles of base 4 m and height 5 m.

3.3 ft2

60 m2

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Areas of Complex ShapesAreas of Complex Shapes

Independent Practice

Solution follows…

Lesson

3.1.4

7. Find the area of the metal bracket shown.

5.45 in2

5 in

4 in2.6 in

3.5 in

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Areas of Complex ShapesAreas of Complex Shapes

Round UpRound Up

Lesson

3.1.4

Take care to include every piece though.

You can find the areas of complex shapes by splitting them up into shapes you know formulas for — squares, rectangles, triangles, trapezoids, parallelograms...