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
5
Divide 360 by # of angles
angle2 = ½ m 1 = 36
angle3 = 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 angles at the center. Use 3060-90 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 mm2
Practice
1. Find the area of a regular pentagon with 11.6 cm sides
and an 8-cm apothem.
P = ns
Area = ½ ap
p = (5)(11.6) = 58
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