Section 11.6 Notes. Regular Polygon and its Parts.
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Section 11.6 Notes
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Transcript of 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
