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Geometry HS Mathematics Unit: 08 Lesson: 01 A Closer Look at Area of Regular Polygons KEY A regular polygon is a convex polygon in which all sides are congruent and all angles are congruent. 108o 60o 108o 60o 60o 108o Equilateral triangle Equiangular triangle Square 108o 108o Regular pentagon Any regular polygon can be divided into non-overlapping, congruent, triangular regions. The area of the regular polygon can be found by finding the area of one of the triangular regions and multiplying by the number of sides of the polygon. This method can be shown to reduce to the formula… 1 aP 2 where P represents the perimeter and a represents the apothem. Area of a regular polygon = A The apothem is a segment that extends from the center of a regular polygon perpendicularly to a side of the regular polygon. The apothem is also the perpendicular distance from the center of the figure to a side. The apothem bisects a side of the regular polygon. In Figure 1 below, a is the length of the apothem. The radius is a segment that extends from the center of the regular polygon to a vertex of the polygon. The radius is also the distance from the center of the polygon to a vertex of the polygon. The radius bisects an interior angle of the regular polygon. In Figure 1 below, r is the length of the radius. 108o 108o 108o a 108o r 54° 54° Attributes of special right triangles, 45°-45°-90°, and 30°-60°-90° and the trigonometric ratios can be used to solve problems involving regular polygons. Geometry HS Mathematics Unit: 08 Lesson: 01 A Closer Look at Area of Regular Polygons KEY EXAMPLES If necessary, round to the nearest hundredth. 1. Find the perimeter and areas of the following regular polygons. a. b. 2.1 in. P=15.5 in. A=16.28 in.2 3.1 in. 6.3 m 7.6 m P=50.4 in. A=191.52 in.2 2. A regular hexagon has an area of approximately 20 square centimeters. If the apothem measures 2.4 centimeters, find the approximate length of each side. Length of each side, s ≈ 2.78 cm. Geometry HS Mathematics Unit: 08 Lesson: 01 A Closer Look at Area of Regular KEY Practice Problems If necessary, round to the nearest tenth. Find the apothem for each of the regular polygons below. Then find the perimeter and area for each figure. 1. 2. 3. 12 cm 15 mm 10 in. a ≈ 4.3 mm P = 45 mm A ≈ 96.8 mm2 a ≈ 8.5 cm P ≈ 67.9 cm A ≈ 288.6 cm2 4. a ≈ 6.9 in. P = 50 in. A ≈ 172.5 in.2 5. 30 in. 6 ft a = 3 3 ft or ≈ (5.2 ft) P = 36 ft. A ≈ 93.6 ft.2 a ≈ 27.7 in. P ≈ 184 in. A ≈ 2548.4 in.2 6. The isosceles trapezoid below is composed of three congruent equilateral triangles with side lengths of 6 cm. Find the area of the trapezoid. area ≈ 46.8 cm2