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.

√ EXAMPLE 2 Find the area of a regular polygon STEP 2 Find the apothem a. The apothem is height RS of ∆PQR. Because ∆PQR is isosceles, altitude RS bisects QP . So, QS = (QP) = (15) = 7.5 inches. 1 2 To find RS, use the Pythagorean Theorem for ∆ RQS. a = RS ≈ √19.62 – 7.52 = 327.91 ≈ 18.108 √

Find the area of a regular polygon EXAMPLE 2 Find the area of a regular polygon 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. So, the area you are covering with tiles is about 1086.5 square inches. ANSWER

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

EXAMPLE 3 Find the perimeter and area of a regular polygon sin 20° = MK LK cos 20° = LM LK sin 20° = MK 4 cos 20° = LM 4 4 sin 20° = MK 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°).

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 ANSWER

GUIDED PRACTICE for Examples 2 and 3 Find the perimeter and the area of the regular polygon. 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. 360 5

GUIDED PRACTICE for Examples 2 and 3 sin 36° = b hyp sin 36° = b 8 8 sin 36° = b The regular pentagon has side length = 2b = 2 (8 sin 36°) = 16 sin 36° 20° So, the perimeter is P = 5s = 5(16 sin 36°) = 80 sin 36° ≈ 46.6 units, and the area is A = aP = 6.5 46.6 1 2 ≈ 151.5 units2.

GUIDED PRACTICE for Examples 2 and 3 Find the perimeter and the area of the regular polygon. 4. SOLUTION The regular nonagon has side length = 7. So, the perimeter is P = 10 · s = 10 · 7 = 70 units

GUIDED PRACTICE for Examples 2 and 3 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. 360 10 tan 18° = opp adj tan 18° = 3.5 a a = 3.5 tan 18° ≈10.8 and the area is A = aP = 10.8 70 1 2 ≈ 377 units2.

GUIDED PRACTICE for Examples 2 and 3 5. 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

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

GUIDED PRACTICE for Examples 2 and 3 So, the perimeter is P = 3 s = 3(20 sin 60°) = 60 sin 60° = 30 3 units and the area is A = aP 1 2 = × 5 30 3 1 2 = 129.9 units2

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 30-60-90 Right Triangle ANSWER