Areas of Regular Polygons Warm Up Lesson Presentation Lesson Quiz

Warm-Up 1. An isosceles triangle has side lengths 20 meters, 26 meters, and 26 meters. Find the length of the altitude to the base. ANSWER 24 m 1 2 2. Solve 18 = x. ANSWER 36

Warm-Up 3. Evaluate (4 cos 10º)(10 sin 20º). ANSWER 13.47 4. Evaluate . 10 tan 88º 0.3492 ANSWER

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

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

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

Guided Practice 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 2. Find m XPY, m XPQ, and m PXQ. 90°, 45°, 45° ANSWER

Example 2 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? DECORATING SOLUTION STEP 1 Find the perimeter P of the table top. An octagon has 8 sides, so P = 8(15) = 120 inches.

Example 2 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 A of the table top. 1 2 A = aP 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. So, the area you are covering with tiles is about 1086.5 square inches.

Example 3 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 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°. 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

Guided Practice Find the perimeter and the area of the regular polygon. 3. about 46.6 units, about 151.6 units2 ANSWER

Guided Practice Find the perimeter and the area of the regular polygon. 4. 70 units, about 377.0 units2 ANSWER

Guided Practice Find the perimeter and the area of the regular polygon. 5. 30 3  52.0 units, about 129.9 units2 ANSWER

Guided Practice 6. Which of Exercises 3–5 above can be solved using special right triangles? Exercise 5 ANSWER

Lesson Quiz 1. Find the measure of the central angle of a regular polygon with 24 sides. ANSWER 15°

Lesson Quiz Find the area of each regular polygon. 2. ANSWER 110 cm2 3. ANSWER 374.1 cm2

Lesson Quiz Find the perimeter and area of each regular polygon. 4. ANSWER 99.4 in. ; 745.6 in.2 5. ANSWER 22.6 m; 32 m2