Areas Of Polygons

GeomPHM5e_DNG_118-141

11/17/2004

2:16 PM

Page 130

Name_____________________________________ Class____________________________ Date________________

Lesson 7-5 (pp. 380–385)

Areas of Regular Polygons

Lesson Objective 1 Find the area of a regular polygon

NAEP 2005 Strand: Measurement Topic: Measuring Physical Attributes Local Standards: ____________________________________

Vocabulary and Key Concepts. Theorem 7-12: Area of a Regular Polygon The area of a regular polygon is

A


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a p

The center of a regular polygon is The radius of a regular polygon is

© Pearson Education, Inc., publishing as Pearson Prentice Hall.

The apothem of a regular polygon is

Examples. 1 Finding Angle Measures This regular hexagon has an apothem and radii drawn. Find the measure of each numbered angle. m1


360

m2





1 2 3

Divide 360 by the number of sides. The apothem bisects the vertex angle of

m1

the

triangle formed by

the radii.

m2


(

)


m3
180 

(

m1


, m2


130

Lesson 7-5



Substitute

)


for m
1. The sum of the measures of the angles of a triangle is

.

, and m3


Geometry Daily Notetaking Guide

GeomPHM5e_DNG_118-141

11/17/2004

2:16 PM

Page 131

Name_____________________________________ Class____________________________ Date ________________

2 Finding the Area of a Regular Polygon A library is in the shape

18.0 ft

of a regular octagon. Each side is 18.0 ft. The radius of the octagon is 23.5 ft. Find the area of the library to the nearest 10 ft 2. Consecutive radii form an isosceles triangle, so an apothem bisects the side of the octagon. To apply the area formula A
12ap, you need to find a and p. Step 1 Find the apothem a. a2 

(

)2
(

a2 

)2




23.5 ft

9.0 ft

9.0 ft

Pythagorean Theorem Solve for a.

a2
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a< Step 2 Find the perimeter p. Find the perimeter, where n  the number of sides of a regular polygon.

p
ns p
(8)

(

)


for n and

for s, and simplify.

Step 3 Find the area A. A
12 ap A  12

(

Area of a

)(

A

)

polygon

Substitute

for a and

for p.

Simplify.

To the nearest 10 ft 2, the area is © Pearson Education, Inc., publishing as Pearson Prentice Hall.

Substitute

ft 2.

3 Applying Theorem 7-12 Find the area of an equilateral triangle with apothem 8 cm. Leave your answer in simplest radical form. This equilateral triangle shows two radii forming an angle that measures s

360
120. Because the radii and a side form an isosceles triangle, 3

the apothem bisects the 120 angle, forming two

angles.

60 8 cm 60

You can use a 30-60-90 triangle to find half the length of a side. 1s 5 2 1s 5 2

?a

longer leg 

?

Substitute

s5 p 5 ns p5

( )(

s

? shorter leg for a.

Multiply each side by

.

Find the perimeter.

)


A 5 12 ap A 5 12

( )(

for n and

for s, and simplify.

Area of a regular polygon

)

A5 The area of the equilateral triangle is

Geometry Daily Notetaking Guide Check Understanding.

Substitute

Substitute

for a and

for p.

Simplify.

cm 2.

Lesson 7-5

131

Name_____________________________________ Class____________________________ Date ________________

Check Understanding. 1. At the right, a portion of a regular octagon has radii and an apothem drawn. Find the measure of each numbered angle. 1 2

2. The side of a regular hexagon is 16 ft. Find the area of the hexagon.

3. Critical Thinking In the figure at right, 16 small equilateral triangles form a large equilateral triangle. Each small triangle has a 12.7-in. radius. What is the area of the large triangle to the nearest square inch?

12.7 in.

132

Lesson 7-5

Geometry Daily Notetaking Guide

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All rights reserved.

3