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Warm Up

Areas of Regular Polygons. Warm Up. Lesson Presentation. Lesson Quiz. ANSWER. 24 m. 1. 2. Solve 18 =. x. 2. 36. ANSWER. 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. 13.47.

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Warm Up

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  1. Areas of Regular Polygons Warm Up Lesson Presentation Lesson Quiz

  2. ANSWER 24 m 1 2. Solve 18 = x. 2 36 ANSWER 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.

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

  4. 360° 5 a. m AFB a. AFB is a central angle,som AFB = , or 72°. Example 1 In the diagram, ABCDEis a regular pentagon inscribed in F. Find each angle measure. SOLUTION

  5. b. m AFG 1 b. FG is an apothem, which makes it an altitude ofisosceles ∆AFB. So, FGbisectsAFB andm AFG = m AFB = 36°. 2 Example 1 In the diagram, ABCDEis a regular pentagon inscribed in F. Find each angle measure. SOLUTION

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

  7. In the diagram, WXYZis a square inscribed in P. 2. Find m XPY, m XPQ, andm PXQ. ANSWER P, PY or XP, PQ, XPY ANSWER 90°, 45°, 45° Guided Practice 1. Identify the center, a radius, an apothem, and a central angle of the polygon.

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

  9. 1 1 So,QS = (QP) = (15) = 7.5 inches. 2 2 √ a = RS ≈ √19.62 – 7.52 = 327.91 ≈ 18.108 Example 2 STEP 2 Find the apothem a. The apothem is height RSof ∆PQR. Because ∆PQRis isosceles, altitude RSbisects QP. To find RS, use the Pythagorean Theorem for ∆ RQS.

  10. 1 A = aP 2 1 ≈ (18.108)(120) 2 Example 2 STEP 3 Find the area Aof the table top. Formula for area of regular polygon Substitute. ≈ 1086.5 Simplify. So, the area you are covering with tiles is about 1086.5square inches.

  11. 360° The measure of central JLKis , or 40°. Apothem LMbisects the central angle, so m KLMis 20°. To find the lengths of the legs, use trigonometric ratios for right ∆KLM. 9 Example 3 A regular nonagon is inscribed in a circle with radius 4 units. Find the perimeter and area of the nonagon. SOLUTION

  12. LM MK sin 20° = cos 20° = LK LK MK LM cos 20° = sin 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 sin20°)≈46.3 square units. 4 4 1 1 4 sin 20° = MK 4 cos 20° = LM 2 2 The regular nonagon has side length s = 2MK = 2(4 sin 20°) = 8  sin 20° and apothem a = LM = 4  cos20°. Example 3

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

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

  15. 5. 30 3  52.0units, about 129.9 units2 ANSWER Guided Practice Find the perimeter and the area of the regular polygon.

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

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

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

  19. 5. 4. ANSWER 99.4 in. ; 745.6 in.2 22.6 m; 32 m2 ANSWER Lesson Quiz Find the perimeter and area of each regular polygon.

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