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Areas of Regular Polygons Lesson 11.5. Equilateral Triangle. Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the sides and altitude in terms of one side equaling s ? Altitude = s √ 3 2. C. Given: ∆ CAT is equilateral, and TA = s

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## Areas of Regular Polygons Lesson 11.5

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**Areas of Regular Polygons**Lesson 11.5**Equilateral Triangle**Remember: drop an altitude and you create two 30-60-90 triangles. What is the measure of the sides and altitude in terms of one side equaling s? Altitude = s√3 2**C**Given: ∆ CAT is equilateral, and TA = s Find the area of ∆CAT A T S A∆CAT = = = 2**Theorem 106: Area of an equilateral triangle = the product**of 1/4 the square of a side and the square root of 3. Where s is the length of a side Aeq∆= 2**An equilateral triangle has a side of 10 cm long. Find the**area of the triangle. A = 102(√3) 4 A = 25√3 cm2**Area of a regular polygon:**Remember all interior angles are congruent and all sides are equal. N Regular pentagon: O is the center OA the radius OM is an apothem T E O P M A**You can make 5 isosceles triangles in a pentagon.**Any regular polygon: Radius: is a segment joining the center to any vertex Apothem: is a segment joining the center to the midpoint of any side.**Apothems:**All apothems of a regular polygon are congruent. Only regular polygons have apothems. An apothem is a radius of a circle inscribed in the polygon. An apothem is the perpendicular bisector of a side. A radius of a regular polygon is a radius of a circle circumscribed about the polygon. A radius of a regular polygon bisects an angle of the polygon.**Theorem 107: Areg poly = ½ ap**Area of a regular polygon equals one-half the product of the apothem and the perimeter. Where a = apothem p = perimeter**A regular polygon has a perimeter of 40 cm and an apothem of**5 cm. Find the polygon’s area. A = ½ap = ½(5)(40) = 100 cm2**Find the area of a regular hexagon whose sides are 18 cm**long. Draw the picture Write the formula Plug in the numbers Solve and label units**Find the perimeter**Find each angle Find the apothem 18cm P = 18(6) = 108 cm Angles = 720º/6 angles = 120º per angle Radius breaks it into 60º angles. 30-60-90 triangle, apothem = 9√3 cm Write the formula, and solve.**A = ½ ap**A = ½ (9√3)108 A = 486√3 cm 2**Team Challenge:**A square is inscribed in an equilateral triangle as shown. Find the area of the shaded region.**2x + x√3 = 12**x = 12 2 + √3 x = 12(2 – √3) A (shaded) = ½ (12)(6√3) – [12(2 – √3)√3]2 = 1764√3 - 3024 x√3 x x√3 x

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