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 PolygonsPerimeter 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.

9

6

R

Q P

S

D

B C

A

Perimeter of Similar PolygonsIf 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.

b

a

IIPolygon ofPerimeter

IPolygon ofPerimeter

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 a2:b2.

Example 11

In the diagram ΔABC ~ ΔDEF. 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.