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EXAMPLE 2 Find the area of a regular polygon DECORATING You are decorating the top of a table by covering it with small ceramic tiles. The table top is a regular octagon with 15 inch sides and a radius of about 19.6 inches. What is the area you are covering? SOLUTION STEP 1 Find the perimeter P of the table top. An octagon has 8 sides, so P = 8(15) = 120 inches.

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EXAMPLE 2 STEP 2 So, QS = (QP) = (15) = 7.5 inches To find RS, use the Pythagorean Theorem for RQS. a = RS – = Find the apothem a. The apothem is height RS ofPQR. BecausePQR is isosceles, altitude RS bisects QP. Find the area of a regular polygon

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EXAMPLE 2 STEP 3 Find the area A of the table top. 1 2 A = aP Formula for area of regular polygon (18.108)(120) 1 2 Substitute Simplify. Find the area of a regular polygon So, the area you are covering with tiles is about square inches. ANSWER

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EXAMPLE 3 Find the perimeter and area of a regular polygon A regular nonagon is inscribed in a circle with radius 4 units. Find the perimeter and area of the nonagon. SOLUTION 360° The measure of central JLK is, or 40°. Apothem LM bisects the central angle, so m KLM is 20°. To find the lengths of the legs, use trigonometric ratios for right KLM. 9

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EXAMPLE 3 sin 20° = MK LK sin 20° = MK 4 4 sin 20° = MK cos 20° = LM LK cos 20° = LM 4 4 cos 20° = LM The regular nonagon has side length s = 2MK = 2(4 sin 20°) = 8(sin 20°) and apothem a = LM = 4(cos 20°). Find the perimeter and area of a regular polygon

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EXAMPLE 3 Find the perimeter and area of a regular polygon So, the perimeter is P = 9s = 9(8 sin 20°) = 72 sin 20° 24.6 units, and the area is A = aP = (4 cos 20°)(72 sin 20°) 46.3 square units ANSWER

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GUIDED PRACTICE for Examples 2 and 3 3. SOLUTION The measure of the central angle is = or 72°. Apothem a bisects the central angle, so angle is 36°. To find the lengths of the legs, use trigonometric ratios for right angle Find the perimeter and the area of the regular polygon.

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GUIDED PRACTICE for Examples 2 and 3 sin 36° = b hyp sin 36° = b 8 So, the perimeter is P = 5s = 5(16 sin 36°) 8 sin 36° = b The regular pentagon has side length = 2b = 2 (8 sin 36°) = 16 sin 36° 20° = 80 sin 36° 46.6 units, and the area is A = aP = units 2.

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GUIDED PRACTICE for Examples 2 and SOLUTION The regular nonagon has side length = 7. So, the perimeter is P = 10 · s = 10 · 7 = 70 units Find the perimeter and the area of the regular polygon.

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GUIDED PRACTICE for Examples 2 and 3 and the area is A = aP = units 2. The measure of central is = or 36°. Apothem a bisects the central angle, so angle is 18°. To find the lengths of the legs, use trigonometric ratios for right angle tan 18° = opp adj tan 18° = 3.5 a a = 3.5 tan 18° 10.8

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GUIDED PRACTICE for Examples 2 and SOLUTION The measure of central angle is = 120°. Apothem a bisects the central angle, so is 60°. To find the lengths of the legs, use the trigonometric ratios. 360° 3

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GUIDED PRACTICE for Examples 2 and 3 cos 60° = a x x = 10 sin 60° = b 10 The regular polygon has side length s = 2 = 2 (10 sin 60°) = 20 sin 60° and apothem a = 5. x cos 60° =5 x 0.5 =5 b 10 sin 60° =

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GUIDED PRACTICE for Examples 2 and 3 = 30 3 units = 60 sin 60° and the area is A = aP 1 2 = × = units 2 So, the perimeter is P = 3 s = 3(20 sin 60°)

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GUIDED PRACTICE for Examples 2 and 3 6. Which of Exercises 3–5 above can be solved using special right triangles? Exercise 5 can be solved using special right triangles. The triangle is a Right Triangle ANSWER

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