3. EXAMPLE 1 Find angle measures in a regular polygon a. m AFB In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure. SOLUTION AFB is a central angle, so m AFB =, or 72 °. 360 ° 5

4. EXAMPLE 1 Find angle measures in a regular polygon b. m AFG In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure. SOLUTION FG is an apothem, which makes it an altitude of isosceles ∆AFB. So, FG bisects AFB and m AFG = m AFB = 36 °. 1 2

5. EXAMPLE 1 Find angle measures in a regular polygon c. m GAF In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure. SOLUTION The sum of the measures of right ∆GAF is 180 °. So, 90 ° + 36 ° + m GAF = 180 °, and m GAF = 54 °.

6. GUIDED PRACTICE for Example 1 In the diagram, WXYZ is a square inscribed in P. 1. Identify the center, a radius, an apothem, and a central angle of the polygon. P, PY or XP, PQ,  XPY. ANSWER

7. GUIDED PRACTICE for Example 1 2. Find m XPY, m XPQ, and m PXQ. 90 °, 45 °, 45 ° ANSWER

9. 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.

10. 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 of ∆PQR. Because ∆PQR is isosceles, altitude RS bisects QP. Find the area of a regular polygon

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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.

18. GUIDED PRACTICE for Examples 2 and 3 6. Which of Exercises 3–5 above can be solved using special right triangles? Exercise 5ANSWER