Lesson 3.1.4

1. Areas of Complex Shapes Lesson 3.1.4

2. 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. 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. 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 thetotal area. 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:

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

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

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

8. = – 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 alarger shape and subtractthe area of a smaller shape.

9. 2 cm 10 cm 5 cm 20 cm Lesson 3.1.4 Areas of Complex Shapes Example 2 Find the shaded area of this shape. 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. 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. Example 3 23 ft 12 ft 7 ft 54 ft Area A = × 54 × 23 = 621 ft2 Area B = × 12 × 7 = 42 ft2 1 1 2 2 Lesson 3.1.4 Areas of Complex Shapes Find the shaded area of this shape. A B 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 Solution follows…

12. 0.7 m 3 m 1 m 3 m Lesson 3.1.4 Areas of Complex Shapes Guided Practice Use subtraction to find the areas of the shapes in Exercises 2–3. 2. 3. 5 ft 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. 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. 2 cm 3 cm 8 cm 5 cm Lesson 3.1.4 Areas of Complex Shapes Independent Practice Use either addition or subtraction to find the areas of the following shapes. 1. 2. 7 mm 6 mm 11 mm 12 mm 27.5 cm2 18 mm2 Solution follows…

15. 20 in 100 ft 40 in 9 in 23 ft 125 ft 50 ft 25 ft 2875 ft2 130 ft Lesson 3.1.4 Areas of Complex Shapes Independent Practice Use either addition or subtraction to find the areas of the following shapes. 3. 4. 580 in2 2875 ft2 Solution follows…

16. 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. 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. 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.