1. 8.4 Areas of Regular Polygons

2. 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

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

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

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

6. 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.

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

8. 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°
=67.5°

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

10. 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

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

12. 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

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

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

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

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

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