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Trigonometry and Area LESSON 10-5 Additional Examples Find the area of a regular polygon with 10 sides and side length 12 cm. Find the perimeter p and apothem a, and then find the area using the formula A = 1 ap. 2 Because the polygon has 10 sides and each side is 12 cm long, p = 10 • 12 = 120 cm. Use trigonometry to find a. Because the polygon has 10 sides, m ACB = 360 = 36. 10 CA and CB are radii, so CA = CB. Therefore, ACM BCM by the HL 1 1 Theorem, so m ACM = 2 m ACB = 18 and AM = 2 AB = 6. HELP GEOMETRY Trigonometry and Area LESSON 10-5 Additional Examples (continued) tan 18° = 6 a Use the tangent ratio. 6 a = tan 18° Solve for a. Now substitute into the area formula. A = 1 ap 2 6 . 1 A = 2 • tan 18° • 120 360 A = tan 18° 360 18 The area is about 1108 cm2. HELP Substitute for a and p. Simplify. Use a calculator. Quick Check GEOMETRY Trigonometry and Area LESSON 10-5 Additional Examples The radius of a garden in the shape of a regular pentagon is 18 feet. Find the area of the garden. Find the perimeter p and apothem a, and then find the area using the formula A = 1 ap. 2 Because the pentagon has 5 sides, m ACB = 360 = 72. 5 CA and CB are radii, so CA = CB. Therefore, HL Theorem, so m ACM = 1 m ACB = 36. ACM BCM by the 2 HELP GEOMETRY Trigonometry and Area LESSON 10-5 Additional Examples (continued) Use the cosine ratio to find a. cos 36° = a 18 a = 18(cos 36°) Use the sine ratio to find AM. Use the ratio. Solve. Use AM to find p. Because ACM pentagon is regular, p = 5 • AB. sin 36° = AM 18 AM = 18(sin 36°) BCM, AB = 2 • AM. Because the So p = 5 • (2 • AM) = 10 • AM = 10 • 18(sin 36°) = 180(sin 36°). HELP GEOMETRY Trigonometry and Area LESSON 10-5 Additional Examples (continued) Finally, substitute into the area formula A = 1 ap. 2 A= 1 • 18(cos 36°) • 180(sin 36°) 2 Substitute for a and p. A = 1620(cos 36°) • (sin 36°) Simplify. A Use a calculator. The area of the garden is about 770 ft2. Quick Check HELP GEOMETRY Trigonometry and Area LESSON 10-5 Additional Examples A triangular park has two sides that measure 200 ft and 300 ft and form a 65° angle. Find the area of the park to the nearest hundred square feet. Use Theorem 10-8: The area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. Area = 1 • side length • side length 2 Theorem 10-8 • sine of included angle Area = 1 • 200 • 300 • sin 65° Substitute. Area = 30,000 sin 65° Simplify. 2 Use a calculator The area of the park is approximately 27,200 ft2. HELP Quick Check GEOMETRY