Area Topic 13: Lesson 3 Composite Figures Holt Geometry Texas ©2007

Objectives and Student Expectations TEKS: G8A The student will find areas of regular polygons, circles, and composite figures.

A composite figure is made up of simple shapes, such as triangles, rectangles, trapezoids, and circles. To find the area of a composite figure, find the areas of the simple shapes and then use the Area Addition Postulate.

Example: 1 Divide the figure into parts. Find the shaded area. Round to the nearest tenth, if necessary. Divide the figure into parts. A = ½ circle + rectangle + triangle

Example: 1 continued A = ½ circle + rectangle + triangle Shaded area: A = ½ circle + rectangle + triangle A = ½  r2 + bh + ½ bh A = ½ ( 102)+ (20)(14) + ½ (12)(14) A = 50 + 280 + 84 A = 50 + 364 mm2 A ≈ 521.1 mm2

Example: 2 Find the shaded area. Divide the figure into parts. A = parallelogram + triangle A = bh + ½ bh A = (8)(5) + ½ (10)(5) A = 40 + 25 A = 65 ft2

Example: 3 Find the shaded area. Find height of triangle: A = rectangle + triangle A = bh + ½ bh A = (37.5)(22.5) + ½ (37.5)(50) A = 843.75 + 937.5 A = 1781.25 m2

Example: 4 A = triangle – ½ circle A = ½ bh – ½  r2 Find the shaded area. Round to the nearest tenth, if necessary. A = triangle – ½ circle A = ½ bh – ½  r2 A = ½ (26)(18) – ½  (4.5)2 A = 234 – 10.125  ft2 A ≈ 202.2 ft2

Example: 5 A = circle – trapezoid A = r2 – ½ (b1 + b2)h Find the shaded area. Round to the nearest tenth, if necessary. A = circle – trapezoid A = r2 – ½ (b1 + b2)h A = (10)2 – ½ (12 + 20)(8) A= 100 – 128 cm2 A  186.2 cm2

Example: 6 A = circle – square A =  r2 - ½ d1d2 A =  32 - ½ (6)(6) Find the shaded area. Round to the nearest tenth, if necessary. Hint: the diagonals of a square are . A = circle – square A =  r2 - ½ d1d2 A =  32 - ½ (6)(6) A = 9 -18 in2 A  10.3 in2

Example: 7 A company receives an order for 65 pieces of fabric in the given shape. Each piece is to be dyed red. To dye 6 in2 of fabric, 2 oz of dye is needed. How much dye is needed for the entire order? Round to the nearest oz. To find the area of the shape in square inches, divide the shape into parts. The two half circles have the same area as one circle.

Example: 7 continued A = circle + square The area of the circle is The area of the square is (3)2 = 9 in2. The total area of the shape is 2.25 + 9 in2 ≈ 16.1 in2. The total area of the 65 pieces is 65(16.1) ≈ 1044.5 in2. The company will need 1044.5 ≈ 348 oz of dye for the entire order.

Example: 8 Area = rectangle + trapezoid + rectangle The lawn that Katie is replacing requires 79 gallons of water per square foot per year. How much water will Katie save by planting the xeriscape garden which only uses 17 gallons of water per square foot per year? Area = rectangle + trapezoid + rectangle A = (28.5)(7.5) + ½ (18 + 12)(6) + (12)(6) A=375.75 ft2 Area times gallons of water saved: 375.75(79 - 17) = 23,296.5 gallons saved.

First, draw a composite figure that resembles the irregular shape. To estimate the area of an irregular shape, you can sometimes use a composite figure. First, draw a composite figure that resembles the irregular shape. rect. triangle triangle trapezoid triangle Then divide the composite figure into simple shapes. Different people will probably see different ways to divide the composite figure—there is not just one way!

Example: 9 Use a composite figure to estimate the shaded area. The grid has squares with side lengths of 1 ft. Draw a composite figure that approximates the irregular shape. Find the area of each part of the composite figure.

Example: 9 continued A = rectangle + triangle + trapezoid + right triangle A = (2)(4) + ½ (2)(2) + ½ (3 + 1)(1) + ½ (2)(1) A = 8 + 2 + 2 + 1 A = 13 ft2