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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 = ap. 1 2. 360 10. Because the polygon has 10 sides, m ACB = = 36.

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Find the area of a regular polygon with 10 sides and side length 12 cm.

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  1. Find the perimeter p and apothem a, and then find the area using the formula A = ap. 1 2 360 10 Because the polygon has 10 sides, mACB = = 36. and are radii, so CA = CB. Therefore, ACMBCM by the HL Theorem, so m ACM = m ACB = 18 and AM = AB = 6. CA CB 1 2 1 2 Trigonometry and Area LESSON 10-5 Additional Examples Find the area of a regular polygon with 10 sides and side length 12 cm. Because the polygon has 10 sides and each side is 12 cm long, p = 10 • 12 = 120 cm. Use trigonometry to find a.

  2. Use the tangent ratio. 6 a tan 18° = 6 tan 18° a = Solve for a. Substitute for a and p. 1 2 A = ap Simplify. 360 18 Use a calculator. 6 . tan 18° 1 2 360 tan 18° A = • • 120 A = Trigonometry and Area LESSON 10-5 Additional Examples (continued) Now substitute into the area formula. The area is about 1108 cm2. Quick Check

  3. Find the perimeter p and apothem a, and then find the area using the formula A = ap. 1 2 360 5 Because the pentagon has 5 sides, mACB = = 72. CA and CB are radii, so CA = CB. Therefore, ACMBCM by the HL Theorem, so mACM = mACB = 36. 1 2 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.

  4. Use the cosine ratio to find a. Use the sine ratio to find AM. AM 18 a 18 sin 36° = cos 36° = Use the ratio. a = 18(cos 36°) Solve. AM = 18(sin 36°) Use AM to find p. Because ACMBCM, AB = 2 • AM. Because the pentagon is regular, p = 5 • AB. Trigonometry and Area LESSON 10-5 Additional Examples (continued) So p = 5 • (2 • AM) = 10 • AM = 10 • 18(sin 36°) = 180(sin 36°).

  5. 1 2 A = • 18(cos 36°) • 180(sin 36°) Substitute for a and p. A = 1620(cos 36°) • (sin 36°) Simplify. Use a calculator. A Trigonometry and Area LESSON 10-5 Additional Examples (continued) 1 2 Finally, substitute into the area formula A = ap. The area of the garden is about 770 ft2. Quick Check

  6. 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. 1 2 Area = • side length • side length Theorem 10-8 • sine of included angle Substitute. 1 2 Area = • 200 • 300 • sin 65° Area = 30,000 sin 65° Simplify. Use a calculator Trigonometry and Area LESSON 10-5 Additional Examples 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. Quick Check The area of the park is approximately 27,200 ft2.

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