31-3: Area and Perimeter of Regular Polygons
Objective:
1.
To find the area and perimeter of a regular n -gon
Assignment:
Purple Geometry Book:
• P. 765-768: 1-7, 10-
18 even, 17, 19, 23-
25, 28, 35, 43, 47, 48
• Challenge Problems

Objective 1
You will be able to find the area and perimeter of a regular n -gon

Example 1
Explain how you could find the area of the regular hexagon shown.

Regular Inscribed Polygon
The diagram shows a regular polygon inscribed in a circle.
Center of circle = center of the polygon
Radius of circle = radius of the polygon

Regular Inscribed Polygon
The apothem of the polygon is the distance from the center to any side of the polygon.
Apothem = height of isosceles triangle with 2 radii as legs

Regular Inscribed Polygon
A central angle of a polygon is an angle formed by two consecutive radii.
Measure of central angle =
360° 𝑛

Example 2
1.
Identify the center, a radius, an apothem, and a central angle of the polygon.
2.
Find 𝑚∠𝑋𝑃𝑌 , 𝑚∠𝑋𝑃𝑄 , 𝑚∠𝑃𝑋𝑄 .

Example 3
Assume a regular n -gon has a side length of s and an apothem of a . Find a formula for the area of the regular n -gon.

Area of a Regular Polygon
The area of a regular n -gon with side length s is half the product of the apothem a and the perimeter P .
𝐴 =
1
2 𝑎 ∙ 𝑃 𝐴 =
1
2 𝑎 ∙ 𝑛 ∙ 𝑠

Regular 3-gon
What is the measure of each central angle in an equilateral triangle?
What is the measure of the angle formed by the apothem and the radius of the triangle?

Regular 4-gon
What is the measure of each central angle in a square?
What is the measure of the angle formed by the apothem and the radius of a square?

Regular 5-gon
What is the measure of each central angle in a regular pentagon? What is the measure of the angle formed by the apothem and the radius of the pentagon?

Regular 6-gon
What is the measure of each central angle in a regular hexagon? What is the measure of the angle formed by the apothem and the radius of the hexagon?

Example 4
Find the area of each regular polygon.

Summary

Example 5
Find the area of each regular polygon.
1.
A = 2.
A = 3.
A =

Example 6
Find the area of each regular polygon.
1.
A = 2.
A =

Example 7
Find the area of a regular octagon with a side length of 6 inches.

Example 8
Find a formula for the area of a regular hexagon in terms of s , the side length.

Example 9
The perimeter of a regular hexagon is 48 cm.
What is the area of the hexagon?

Example 10
Find the area of the shaded region.

31-3: Area and Perimeter of Regular Polygons
Objective:
1.
To find the area and perimeter of a regular n -gon
Assignment:
Purple Geometry Book:
• P. 765-768: 1-7, 10-
18 even, 17, 19, 23-
25, 28, 35, 43, 47, 48
• Challenge Problems