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

Areas of Regular Polygons
Perimeter and Area of Similar Figures
Objective:
1.
To find the area of a regular n -gon
2.
To describe the effects on perimeter and area when dimensions are changed proportionally

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 .

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

Example 8
The perimeter of a regular hexagon is 48 cm.
What is the area of the hexagon?

Example 9
Find the area of the shaded region.

Example 10
Rectangle ABCD ~ PQRS with a scale factor of 3:4.
Find the perimeter and area of rectangle PQRS .
A 9
D
Q
6
B C R
P
S

Perimeter of Similar Polygons
If two polygons are similar with the lengths of corresponding sides in the ratio of a : b , then the ratio of their perimeters is a : b .
Perimeter of Polygon I
Perimeter of Polygon II
 a b

Area of Similar Polygons
If two polygons are similar with the lengths of corresponding sides in the ratio of a : b , then the ratio of their areas is a 2 : b 2 .

Example 11
In the diagram Δ ABC ~ Δ DEF . Find the indicated ratio.
1.
Ratio (red to blue) of the perimeters
2.
Ratio (red to blue) of the areas

Example 12
Stuart is installing the same carpet in a bedroom and den.
The floors of the rooms are similar. The carpet for the bedroom costs
$117. Carpet is sold by the square foot.
How much does it cost to carpet the den?

Example 13
The polygons below are similar. Find the values of x and y .