Download 11.6 Areas of Regular Polygons

11.6
Areas of Regular Polygons
Goal
Your Notes
p Find areas of regular polygons inscribed in circles.
VOCABULARY
Center of a polygon
Radius of a polygon
Apothem of a polygon
Central angle of a regular polygon
Example 1
Find angle measures in a regular polygon
In the diagram, ABCDEF is a regular
hexagon inscribed in (G. Find each
angle measure.
C
B
D
G
E
H
a. m∠EGF
A
b. m∠EGH
F
c. m∠HEG
Solution
a. ∠EGF is a central angle, so m∠EGF 5
, or
}
b. GH is an apothem, which makes it an
}
of isosceles nEGF. So, GH
∠EGF and
m∠EGH 5
m∠EGF 5
.
.
c. The sum of the measures of right nHEG is 1808.
So,
1
1 m∠HEG 5 1808, and
m∠HEG 5
.
316
Lesson 11.6 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
11.6
Areas of Regular Polygons
Goal
Your Notes
p Find areas of regular polygons inscribed in circles.
VOCABULARY
Center of a polygon The center of a polygon is the
center of its circumscribed circle.
Radius of a polygon The radius of a polygon is the
radius of its circumscribed circle.
Apothem of a polygon The distance from the center
to any side of the polygon is the apothem.
Central angle of a regular polygon A central angle of
a regular polygon is an angle formed by two radii
drawn to consecutive vertices of the polygon.
Example 1
Find angle measures in a regular polygon
In the diagram, ABCDEF is a regular
hexagon inscribed in (G. Find each
angle measure.
C
B
D
G
E
H
a. m∠EGF
A
b. m∠EGH
F
c. m∠HEG
Solution
3608
a. ∠EGF is a central angle, so m∠EGF 5 } , or 608 .
6
}
b. GH is an apothem, which makes it an altitude
}
of isosceles nEGF. So, GH bisects ∠EGF and
1
m∠EGH 5 } m∠EGF 5 308 .
2
c. The sum of the measures of right nHEG is 1808.
So, 908 1 308 1 m∠HEG 5 1808, and
m∠HEG 5 608 .
316
Lesson 11.6 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
THEOREM 11.11: 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, so
1
A5}
2
1
, or A 5 }
2
p
a
s
.
Find the area of a regular polygon
Example 2
Coaster A wooden coaster is a regular
octagon with 3 centimeter sides and
a radius of about 3.92 centimeters.
What is the area of the coaster?
3 cm
R
3.92 cm
P
Solution
Step 1 Find the perimeter P of the coaster. An octagon
sides, so P 5
(
)5
centimeters.
has
Step 2 Find the apothem a. The apothem is
height
of nPQR. Because nPQR
}
}
is isosceles, altitude RS
QP.
So, QS 5
(QP) 5
(
R
)
P
5
S
cm.
1.5 cm
To find RS, use the Pythagorean Theorem
for nRQS.
a 5 RS
}}
<Ï
In general, your
answer will be
more accurate if
you avoid rounding
until the last
step. Round your
final answers to
the nearest tenth
unless you are told
otherwise.
2
2
2
}
5Ï
<
Step 3 Find the area A of the coaster.
1
Formula for area
of regular polygon
A5}
aP
2
1
(
<}
2
<
)(
)
Substitute.
Simplify.
The area of the coaster is about
centimeters.
Copyright © Holt McDougal. All rights reserved.
square
Lesson 11.6 • Geometry Notetaking Guide
317
Your Notes
THEOREM 11.11: 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, so
a
1
1
} a p ns .
aP
,
or
A
5
A5}
2
2
s
Find the area of a regular polygon
Example 2
Coaster A wooden coaster is a regular
octagon with 3 centimeter sides and
a radius of about 3.92 centimeters.
What is the area of the coaster?
3 cm
R
3.92 cm
P
Solution
Step 1 Find the perimeter P of the coaster. An octagon
has 8 sides, so P 5 8 ( 3 ) 5 24 centimeters.
Step 2 Find the apothem a. The apothem is
height RS of nPQR. Because nPQR
}
}
is isosceles, altitude RS bisects QP.
R
1
1
So, QS 5 } (QP) 5 } ( 3 )
2
2
P
5 1.5 cm.
S
1.5 cm
To find RS, use the Pythagorean Theorem
for nRQS.
a 5 RS
}}
< Ï 3.92
In general, your
answer will be
more accurate if
you avoid rounding
until the last
step. Round your
final answers to
the nearest tenth
unless you are told
otherwise.
2
2 1.5
2
}
5 Ï 13.1164 < 3.622
Step 3 Find the area A of the coaster.
1
Formula for area
of regular polygon
A5}
aP
2
1
( 3.622 )( 24 )
<}
2
Substitute.
< 43.5
Simplify.
The area of the coaster is about 43.5 square
centimeters.
Copyright © Holt McDougal. All rights reserved.
Lesson 11.6 • Geometry Notetaking Guide
317
Your Notes
Checkpoint Complete the following exercises.
1. In the diagram, FGHJ is a square
inscribed in (K. Find m∠FKJ and
m∠KJF.
G
H
K
L
F
2. The radius of the regular pentagon
is about 6.8 inches. Find the area
to the nearest square inch.
J
8 in.
R
6.8 in.
T
S
Find the perimeter and area of a regular polygon
Example 3
A regular nonagon is inscribed in a
circle with radius 5 units. Find the
perimeter P and area A of the nonagon.
L
K
5
5 M
J
, or
.
The measure of central ∠JLK is
}
Apothem LM bisects the central angle, so m∠KLM
is
. To find the lengths of the legs, use trigonometric
ratios for right nKLM.
sin
MK
5}
5
LM
5}
5
cos
5 MK
L
208
5 LM
5
The regular nonagon has side lengths
s 5 2MK 5 2(
)5
and apothem a 5 LM 5
.
J
5
M
K
So,
P 5 9s 5 9(
)5
<
units,
and
1
A5}
aP
2
1
5}
(
2
318
Lesson 11.6 • Geometry Notetaking Guide
)(
)<
square units.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Complete the following exercises.
1. In the diagram, FGHJ is a square
inscribed in (K. Find m∠FKJ and
m∠KJF.
G
H
K
m∠FKJ 5 908; m∠KJF 5 458
L
F
2. The radius of the regular pentagon
is about 6.8 inches. Find the area
to the nearest square inch.
8 in.
R
6.8 in.
about 110 in.2
Example 3
J
T
S
Find the perimeter and area of a regular polygon
A regular nonagon is inscribed in a
circle with radius 5 units. Find the
perimeter P and area A of the nonagon.
L
K
5
5 M
J
3608
The measure of central ∠JLK is } , or 408 .
9
}
Apothem LM bisects the central angle, so m∠KLM
is 208 . To find the lengths of the legs, use trigonometric
ratios for right nKLM.
MK
LM
sin 208 5 }
5
cos 208 5 }
5
5 sin 208 5 MK
5 cos 208 5 LM
The regular nonagon has side lengths
s 5 2MK 5 2( 5 sin 208 ) 5 10 sin 208
and apothem a 5 LM 5 5 cos 208 .
L
208
5
J
5
M
K
So,
P 5 9s 5 9( 10 sin 208 ) 5 90 sin 208 < 30.8 units,
and
1
A5}
aP
2
1
5}
( 5 cos 208 )( 90 sin 208 ) < 72.3 square units.
2
318
Lesson 11.6 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Complete the following exercise.
3. Find the perimeter and area of the
equilateral triangle inscribed in (F.
F
6
H
6
J
G
FINDING LENGTHS IN A REGULAR N-GON
To find the area of a regular n-gon with radius r,
you may need to first find the apothem a or the
side length s.
You can use . . .
. . . when you know
n and . . .
. . . as in . . .
Pythagorean
Theorem:
Two measures:
r and a, or r and s
Example 2 and
Checkpoint Ex. 2
Special Right
Triangles
Any one measure:
r or a or s
And the value of n
is 3, 4, or 6
Checkpoint Ex. 3
Trigonometry
Any one measure:
r or a or s
Example 3 and
Checkpoint Ex. 3
1 }12 s 2
2
Homework
Copyright © Holt McDougal. All rights reserved.
1 a2 5 r 2
Lesson 11.6 • Geometry Notetaking Guide
319
Your Notes
Checkpoint Complete the following exercise.
3. Find the perimeter and area of the
equilateral triangle inscribed in (F.
Perimeter: about 31.2 units;
Area: about 46.8 square units
F
6
H
6
J
G
FINDING LENGTHS IN A REGULAR N-GON
To find the area of a regular n-gon with radius r,
you may need to first find the apothem a or the
side length s.
You can use . . .
. . . when you know
n and . . .
. . . as in . . .
Pythagorean
Theorem:
Two measures:
r and a, or r and s
Example 2 and
Checkpoint Ex. 2
Special Right
Triangles
Any one measure:
r or a or s
And the value of n
is 3, 4, or 6
Checkpoint Ex. 3
Trigonometry
Any one measure:
r or a or s
Example 3 and
Checkpoint Ex. 3
1 }12 s 2
2
Homework
Copyright © Holt McDougal. All rights reserved.
1 a2 5 r 2
Lesson 11.6 • Geometry Notetaking Guide
319