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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. 1 2 1 2 To find RS, use the Pythagorean Theorem for RQS. a = RS 19.6 2 – 7.5 2 = 327.91 18.108 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. 1086.5 Simplify. Find the area of a regular polygon So, the area you are covering with tiles is about 1086.5 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. 1 2 1 2 ANSWER

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GUIDED PRACTICE for Examples 2 and 3 3. Find the perimeter and the area of the regular polygon. about 46.6 units, about 151.5 units 2 ANSWER

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GUIDED PRACTICE for Examples 2 and 3 4.4. Find the perimeter and the area of the regular polygon. 70 units, about 377.0 units 2 ANSWER

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GUIDED PRACTICE for Examples 2 and 3 5.5. 30 3 52.0 units, about 129.9 units 2 ANSWER Find the perimeter and the area of the regular polygon.

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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 5ANSWER

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FeatureLesson Geometry Lesson Main 1.For the similar rectangles, give the ratios (smaller to larger) of the perimeters and of the areas. 2.The triangles.

FeatureLesson Geometry Lesson Main 1.For the similar rectangles, give the ratios (smaller to larger) of the perimeters and of the areas. 2.The triangles.

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