Presentation on theme: "EXAMPLE 2 Find the area of a regular polygon DECORATING"— Presentation transcript:
1EXAMPLE 2Find the area of a regular polygonDECORATINGYou 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?SOLUTIONSTEP 1Find the perimeter P of the table top. Anoctagon has 8 sides, so P = 8(15) = 120 inches.
2√ EXAMPLE 2 Find the area of a regular polygon STEP 2 Find the apothem a. The apothem is heightRS of ∆PQR. Because ∆PQR is isosceles,altitude RS bisects QP .So, QS = (QP) = (15) = 7.5 inches.12To find RS, use the PythagoreanTheorem for ∆ RQS.a = RS ≈ √19.62 – 7.52 = ≈√
3Find the area of a regular polygon EXAMPLE 2Find the area of a regular polygonSTEP 3Find the area A of the table top.12A = aPFormula for area of regular polygon≈ (18.108)(120)12Substitute.≈Simplify.So, the area you are covering with tiles is aboutsquare inches.ANSWER
4EXAMPLE 3Find the perimeter and area of a regular polygonA regular nonagon is inscribed in a circle with radius 4 units. Find the perimeter and area of the nonagon.SOLUTION360°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
5EXAMPLE 3Find the perimeter and area of a regular polygonsin 20° =MKLKcos 20° =LMLKsin 20° =MK4cos 20° =LM44 sin 20° = MK4 cos 20° = LMThe regular nonagon has side lengths = 2MK = 2(4 sin 20°) = 8(sin 20°)and apothem a = LM = 4(cos 20°).
6EXAMPLE 3Find the perimeter and area of a regular polygonSo, 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.12ANSWER
7GUIDED PRACTICEfor Examples 2 and 3Find the perimeter and the area of the regular polygon.3.SOLUTIONThe 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.3605
8GUIDED PRACTICEfor Examples 2 and 3sin 36° =bhypsin 36° =b88 sin 36° = bThe regular pentagon has side length = 2b = 2(8 sin 36°) = 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 =12≈ units2.
9GUIDED PRACTICEfor Examples 2 and 3Find the perimeter and the area of the regular polygon.4.SOLUTIONThe regular nonagon has side length = 7.So, the perimeter is P = 10 · s = 10 · 7 = 70 units
10GUIDED PRACTICEfor Examples 2 and 3The 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.36010tan 18° =oppadjtan 18° =3.5aa =3.5tan 18°≈10.8and the area is A = aP =12≈ 377 units2.
11GUIDED PRACTICEfor Examples 2 and 35.SOLUTIONThe 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
12GUIDED PRACTICEfor Examples 2 and 3cos 60° =axsin 60° =b10x cos 60° =5b10 sin 60° =x =5x =10The regular polygon has side lengths = 2 = 2 (10 sin 60°) = 20 sin 60° and apothem a = 5.
13GUIDED PRACTICEfor Examples 2 and 3So, the perimeter is P = 3 s = 3(20 sin 60°)= 60 sin 60°= unitsand the area is A = aP12= ×12= units2
14GUIDED PRACTICEfor Examples 2 and 36. Which of Exercises 3–5 above can be solvedusing special right triangles?Exercise 5 can be solved using special right triangles.The triangle is a Right TriangleANSWER