Section 7-5 Areas of Regular Polygons SPI 21B: solve equations to find length, width, perimeter and areaSPI 32L: determine the area of indicated regions involving figuresSPI 41A: determine the perimeter & area of 3 or 4 sided plane figures

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.

m1 = 360 = 72 Divide 360 by # of angles 5

m2 = ½ m1 = 36

m3 = 180 – (90 + 36) = 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 in2

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 = 53

2. Find perimeter: p = ns = (6)(10) = 60

3. Find Area A = ½ ap = ½ (53)(60) 259.8 mm2

Practice

1. Find the area of a regular pentagon with 11.6 cm sides and an 8-cm apothem.

P = ns p = (5)(11.6) = 58Area = ½ ap A = ½ (8)(58) = 232 cm2

2. The side of a regular hexagon is 16 ft. Find the area.

a = 83 (30-60-90 triangle)p = ns = (6)(16) = 96A = ½ ap = ½ (83)(96) = 3843