Section 10-3 Areas of Regular Polygons Objectives: find area of a regular polygon Regular Polygon: equilateral and equiangular

Parts of Regular Polygons Circle is circumscribed about the polygon Radius: - distance from center to vertex - divides figure into n congruent isosceles triangles Apothem: perpendicular distance from the center to the side of polygon

Finding Angle Measures of Regular Polygons The figure at the right is a regular polygon. Find the measure of each numbered angle. angle1 = 360 = 72 Divide 360 by # of angles 5 angle2 = ½ m  1 = 36 angle3 = 180 – ( ) = 54

Area of a Regular Polygon Regular Polygon: all sides and angles are ≅ Radii: divides the figure into ≅ isosceles ∆ Area of Triangle = ½ bh or ½ as There are n ≅ sides and triangles, so: Area of n-gon = n ∙ ½ as or ½ ans Perimeter (p) = ns Using substitution: A = ½ ap

Find Area of a Regular Polygon Find the area of a regular decagon with 12.3 apothem and 8 in sides. 1. Find the perimeter: p = ns = (10)(8) = 80 in 2. Use formula for area of regular polygon: A = ½ ap = ½ (12.3)(80) = 492 in 2

Real-world and Regular Polygons Some boats used for racing have bodies made of a honeycomb of regular hexagonal prisms sandwiched between layers of outer material. At the right is one of those cells. Find its area. The radii form six 60 degree angles at the center. Use triangle to find apothem. long leg = short ∙ √3 1. Find apothem: a = 5√3 2. Find perimeter: p = ns = (6)(10) = Find Area A = ½ ap = ½ (5√3)(60) ≈259.8 mm 2

Practice 1. Find the area of a regular pentagon with 11.6 cm sides and an 8-cm apothem. P = nsp = (5)(11.6) = 58 Area = ½ ap A = ½ (8)(58) = 232 cm2 2. The side of a regular hexagon is 16 ft. Find the area. a = 8√3 ( triangle) p = ns = (6)(16) = 96 A = ½ ap = ½ (8√3)(96) = 384√3