8.4 Areas of Regular Polygons

Area of Regular Polygons A regular polygon has equal sides and equal angles The radius of the polygon is the distance from the center to a vertex radius center

Area of Regular Polygons Drawing all of the radii will create several congruent, isosceles triangles.

Area of Regular Polygons The apothem is the perpendicular distance from the center to the midpoint of one of the sides. center apothem

Area of Regular Polygons An apothem bisects one of the angles created by two radii, creating another pair of congruent triangles radii apothem

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.

Area of Regular Polygons Example: Find the measure of each numbered angle m∠1= m∠2= m∠3= 360∕6 = 60° 60∕2 = 30° 180-90-30=60° 1 2 3

Area of Regular Polygons Example: Find the measure of each numbered angle 6 4 5 m∠4= m∠5= m∠6= 360∕8 = 45° 45∕2 = 22.5° 180-90-22.5=67.5°

Area of Regular Polygons Example: Find the measure of each numbered angle m∠7= m∠8= m∠9= 360∕5= 72° 72∕2 = 36° 180-90-36=54° 7 8 9

Area of Regular Polygons The area of a regular polygon equals one half the product of the apothem and the perimeter or Perimeter equals the side length times the number of sides: p = s*n A= ½ asn

Area of Regular Polygons Example: Find the area of the regular polygon 8 in 12.3 in

Area of Regular Polygons Example: Find the area of the regular polygon Use Pythagorean Theorem! 18 ft 23.5 ft 21.7 ft 9 ft

Area of Regular Polygons Example: Find the area of the regular polygon Use Pythagorean Theorem! 6.1 4.9 3.6 7.2

Area of Regular Polygons Example: Find the area of the regular polygon Use shortcut for 30-60-90 triangle! 10 cm 30 10 5√3 60 5

Area of Regular Polygons Example: Find the area of the regular polygon Use shortcut for 30-60-90 triangle! 60 4.5 15.6 m 30 7.8

Team Challenge: A square is inscribed in an equilateral triangle as shown. Find the area of the shaded region.

A (shaded) = ½ (12)(6√3) – [12(2 – √3)√3]2 = 1764√3 - 3024 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