1. Areas of Complex Shapes
Lesson 3.1.4

2. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
California Standard:
Measurement and Geometry 2.2
Estimate 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

3. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
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.

4. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Complex Shapes Can Be Broken into Parts
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:
Break it up into shapes that you know how to find the area of.
Find the area of each part separately.
Add the areas of each part together to get the total area.

5. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Shapes can often be broken up in different ways. Whichever way you choose, you’ll get the same total area.

6. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Example 1
5 cm
Find the area of this shape. A
2 cm
1.8 cm
4 cm
Solution B
3.2 cm
Split the shape into a rectangle and a triangle.
Area A is a rectangle. Area A = bh = 5 cm × 2 cm = 10 cm2.
Area B is a triangle.
Area B = bh = × 2 cm × 1.8 cm = 1.8 cm2. 1 2
Total area = area A + area B = 10 cm cm2 = 11.8 cm2
Solution follows…

7. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Guided Practice
1. Find the area of the complex shape shown.
50 ft
= 1356 ft2
50 • 18 = 900 ft2
32 • 12 = 384 ft2
12 • 6 = 72 ft2
12 ft
12 ft
18 ft
30 ft
18 ft
12 ft
32 ft
Solution follows…

8. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
You Can Find Areas By Subtraction Too
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. = –

9. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Example 2
Find the shaded area of this shape.
2 cm
5 cm
20 cm
10 cm B A
Solution
First calculate the area of rectangle A, then subtract the area of rectangle 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
Solution follows…

10. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
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.

11. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Example 3
Find the shaded area of this shape.
54 ft
23 ft
12 ft
7 ft A B
Solution
First calculate the area of triangle A, then subtract the area of triangle B.
Area A = × 54 × 23 = 621 ft2 1 2
Area B = × 12 × 7 = 42 ft2 1 2
Total area = area A – area B = 621 ft2 – 42 ft2 = 579 ft2
Solution follows…

12. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Guided Practice
Use subtraction to find the areas of the shapes in Exercises 2–3.
5 ft
0.7 m
1 m
3 m
4 ft
5 ft
7 ft
(3 • 3.7) – (0.7 • 1) = 10.4 m2
(5 • 7) – (0.5 • 5 • 4) = 25 ft2
Solution follows…

13. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Guided Practice
Use subtraction to find the areas of the shape in Exercise 4. 4.
Use subtraction to find the areas of the shape in Exercise 4. 4.
100 in
100 in
30 in
30 in
20 in
(100 • 30) – (30 • 20) = 2400 in2
Solution follows…

14. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Independent Practice
Use either addition or subtraction to find the areas of the following shapes.
8 cm
5 cm
3 cm
2 cm
6 mm
7 mm
11 mm
12 mm
27.5 cm2
18 mm2
Solution follows…

15. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Independent Practice
Use either addition or subtraction to find the areas of the following shapes.
2875 ft2
100 ft
125 ft
130 ft
50 ft
23 ft
25 ft
40 in
20 in
9 in
580 in2
2875 ft2
Solution follows…

16. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Independent Practice
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
Solution follows…

17. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Independent Practice
7. Find the area of the metal bracket shown.
5.45 in2
4 in
2.6 in
3.5 in
5 in
Solution follows…

18. Areas of Complex Shapes
Lesson
3.1.4
Areas of Complex Shapes
Round Up
You can find the areas of complex shapes by splitting them up into shapes you know formulas for — squares, rectangles, triangles, trapezoids, parallelograms...
Take care to include every piece though.