Section 11.6 Notes. Regular Polygon and its Parts.

Section 11.6 Notes

Regular Polygon and its Parts

A regular polygon is an equilateral and equiangular polygon.

E C

D

BA

The center of a regular polygon is the center of the circumscribed circle.

Pt. O is the center of regular pentagon ABCDE.

A B

E C

D

O

The radius of a regular polygon is a segment connecting its center with a vertex.

is a radius of

regular pentagon

.

OB

ABCDE

A B

E C

D

O

How many radii does a regular pentagon have?

5 radii

How many radii will any regular polygon have?

The number of sides

An apothem of a regular polygon is the distance from the center to the side of the regular polygon.

is an apothem

of regular pent.

.

OF

ABCDE

A B

E C

D

O

F

A central angle of a regular polygon is an angle whose vertex is the center of the polygon and whose sides contains consecutive vertices of the polygon.

is a central angle

of regular pentagon .

AOB

ABCDE

A B

E C

D

O

1. Draw a regular triangle, a regular quadrilateral, and a regular hexagon.

2. Draw all the central angles of each of the regular polygons.

3. What must be true of all central angles of a regular polygon and what is the sum of these angles?

They are congruent.

Their sum is 360 .

4. Find the measure of one central angle of each of the regular polygons from #1.

regular triangle is 120°

regular quadrilateral is 90°

regular hexagon is 60°

5. Find measure of one central angle of a regular n-gon.

measure of one central angle

360of a regular gon = n

n

Example 1

Find the measure of a central angle of the following regular polygons:

a) a regular pentagon

36072

5

c) a regular 20-gon

b) a regular octagon

36045

8

36018

20

Find the apothem and radius of each of the following regular polygons.

Example 2

1. a regular triangle with a side length of 12

12

ar

30

60

6

6 3

6 3

3 3

6 3

3

2 3

a

a

2 2 3 4 3 r

2. a regular quadrilateral with a side length of 20

10a

10 2r

45ar 45

20

10

3. a regular hexagon with a side length of 8.

4 3a

8r

60

ar

8

4

Area of a Regular Polygon

The area of a regular polygon is half the product of the apothem and the perimeter of the regular polygon.

1

2A aP

A B

E C

D

O

a s

Example 3Find the area of each of the following regular polygons.

2

13 3 54

2

81 3 in

A

1. a regular triangle with a side length of 18 in.

18 in.

a30

9

93 3

3 a

3 18 54 P

2. a regular quadrilateral with a side length of 12 ft. using the area of a regular polygon formula

45a

12 ft.

6

6a

4 12 48 P

2

16 48

2

144 ft.

A

2

13 3 36

2

54 3 cm

A

3. a regular hexagon with a side length of 6 cm

60

a

6 cm.

3

3 3a

6 6 36 P

Example 4Find the area of a regular octagon with side length of 14 cm to the nearest centimeter.

14 cm

722.5

a

tan 22.57

a

7 tan 22.5 a2.9a

8 14 112 P

212.9 112 162 cm.

2 A

Example 5

Find the area of a regular pentagon with side length of 12 cm. to the nearest hundredth.

a

12 cm.

654

tan546

a

6tan54a

8.258a

5 12 60 P

218.258 60 247.74 cm

2 A