1. Section 7-5 Areas of Regular Polygons SPI 21B: solve equations to find length, width, perimeter and area SPI 32L: determine the area of indicated regions involving figures SPI 41A: determine the perimeter & area of 3 or 4 sided plane figures Objectives: find area of a regular polygon Regular Polygon: equilateral and equiangular

2. 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

3. Finding Angle Measures of Regular Polygons The figure at the right is a regular polygon. Find the measure of each numbered angle. m  1 = 360 = 72 Divide 360 by # of angles 5 m  2 = ½ m  1 = 36 m  3 = 180 – (90 + 36) = 54

4. 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

5. 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

6. 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  s at the center. Use 30-60- 90 triangle to find apothem. long leg = short ∙  3 1. Find apothem: a = 5  3 2. Find perimeter: p = ns = (6)(10) = 60 3. Find Area A = ½ ap = ½ (5  3)(60)  259.8 mm 2

7. 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 (30-60-90 triangle) p = ns = (6)(16) = 96 A = ½ ap = ½ (8  3)(96) = 384  3