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1 Lesson 3.1.4 Areas of Complex Shapes Areas of Complex Shapes

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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: Youll 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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3 Areas of Complex Shapes Lesson 3.1.4 Youve practiced finding the areas of regular shapes. Now youre going to use what youve learned to find areas of more complex shapes.

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

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6 B A Areas 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 cm 2. Total area = area A + area B = 10 cm 2 + 1.8 cm 2 = 11.8 cm 2 5 cm 2 cm 4 cm 3.2 cm 1.8 cm Area B is a triangle. Area B = bh = × 2 cm × 1.8 cm = 1.8 cm 2. 1 2 1 2

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7 50 ft 30 ft 12 ft 18 ft 12 ft Areas 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 ft 2 50 18 = 900 ft 2 32 12 = 384 ft 2 12 6 = 72 ft 2

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8 Areas of Complex Shapes You Can Find Areas By Subtraction Too Lesson 3.1.4 So far weve looked at complex shapes where you add together the areas of the different parts. For some shapes, its easiest to find the area of a larger shape and subtract the area of a smaller shape. =–

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9 Areas 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 cm 2 Area B = lw = 5 × 2 = 10 cm 2 Total area = area A – area B = 200 cm 2 – 10 cm 2 = 190 cm 2

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10 Areas of Complex Shapes Lesson 3.1.4 Since there are many stages to these questions, always explain what youre 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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11 Areas 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 ft 2 – 42 ft 2 = 579 ft 2 Area B = × 12 × 7 = 42 ft 2 1 2 Area A = × 54 × 23 = 621 ft 2 1 2 54 ft 23 ft 12 ft 7 ft A B

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12 Areas 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 3.7) – (0.7 1) = 10.4 m 2 (5 7) – (0.5 5 4) = 25 ft 2 5 ft 7 ft 5 ft 4 ft

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13 Areas 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 in 2 100 in Use subtraction to find the areas of the shape in Exercise 4. 4. 100 in 30 in 20 in 30 in

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14 Areas 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 cm5 cm 3 cm 2 cm 12 mm 11 mm 6 mm 7 mm 27.5 cm 2 18 mm 2

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15 Areas 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 in 2 40 in 20 in 9 in 2875 ft 2 100 ft 125 ft 130 ft 50 ft 23 ft 25 ft 2875 ft 2

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16 Areas 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 ft 2 60 m 2

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17 Areas of Complex Shapes Independent Practice Solution follows… Lesson 3.1.4 7. Find the area of the metal bracket shown. 5.45 in 2 5 in 4 in 2.6 in 3.5 in

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18 Areas of Complex Shapes Round 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...

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