Download 11–3 Areas of Regular Polygons and Circles

11– 3
Areas of Regular Polygons and Circles
BUILD YOUR VOCABULARY
MAIN IDEAS
(page 278)
center
An apothem is a segment that is drawn from the
• Find areas of regular
polygons.
of a regular polygon
• Find areas of circles.
KEY CONCEPT
Area of a Regular
Polygon If a regular
polygon has an area
of A square units, a
perimeter of P units,
and an apothem of a
1
units, then A = _
Pa.
Write the
formula for the area of a
regular polygon under
the tab for Lesson 11-3.
to a side of the
polygon.
Area of a Regular Polygon
"
Find the area of a regular
pentagon with a perimeter
of 90 meters.
!
Apothem:
The central angles of a regular
pentagon are all congruent.
Therefore, the measure of each
−−−
360
angle is _
or 72. GF is an
#
'
5
%
&
apothem of pentagon ABCDE. It
−−−
bisects ∠EGD and is a perpendicular bisector of ED. So,
$
1
(72) or 36. Since the perimeter is 90 meters, each
m∠DGF = _
2
side is 18 meters and FD = 9 meters.
−−
Write a trigonometric ratio to find the length of GF.
length of opposite side
tan θ = __
DF
tan∠DGF = _
length of adjacent side
GF
tan
36°
9
=_
GF
m∠DGF =
DF =
(GF)tan
36°
=
9
9
GF ≈
280
Glencoe Geometry
9
Multiply each side by GF.
GF = __
tan
36 ,
Divide each side by tan 36° .
36°
12.4
Use a calculator.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
perpendicular
11–3
1
Pa
Area: A = _
Area of a regular polygon
2
≈
1
_
(90)(12.4)
90 , a ≈
P=
2
≈ 558
12.4
Simplify.
The area of the pentagon is about
558
square meters.
2
Check Your Progress
Find the area
of a regular pentagon with a perimeter
of 120 inches.
6
3
-
about 991 in2
5
.
4
Use Area of a Circle to Solve a Real-World Problem
KEY CONCEPT
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Area of a Circle If a circle
has an area of A square
units and a radius of
r units, then A = πr 2.
MANUFACTURING An outdoor
accessories company
manufactures circular covers
for outdoor umbrellas. If the
cover is 8 inches longer than
the umbrella on each side,
find the area of the cover in
square yards.
nʈ˜°
ÇÓʈ˜°
The diameter of the umbrella is 72 inches, and the cover must
extend 8 inches in each direction. So the diameter of the cover
8
is
+ 72
+
8
or 88
inches. Divide by 2 to find
that the radius is 44 inches.
A = πr2
= π (44)2
≈
6082.1
Area of a circle
Substitution
Use a calculator.
The area of the cover is
6082.1 square inches. To
convert to square yards, divide by 1296. The area of the
cover is 4.7 square yards to the nearest tenth.
Check Your Progress
A swimming pool
company manufactures circular covers
for above-ground pools. If the cover is
10 inches longer than the pool on each side,
find the area of the cover in square yards.
Ó£Èʈ˜°
£äʈ˜°
33.8 yd2
Glencoe Geometry
281
11–3
Area of an Inscribed Polygon
REVIEW IT
Find the area of the shaded region.
Assume that the triangle is equilateral.
Draw a 30°-60°-90°
triangle with the shorter
leg labeled 5 meters
long. Label the angles
and the remaining sides.
(Lesson 8-3)
The area of the shaded region is the
difference between the area of the circle
and the area of the triangle. First, find
the area of the circle.
A = πr2
Area of a circle
= π (7)2
Substitution
153.9
≈
ÇÊV“
Use a calculator.
8
To find the area of the triangle, use
properties of 30°-60°-90° triangles.
First, find the length of the base. The
hypotenuse of RSZ is 7, so RS is 3.5
and SZ is 3.5 √
3 . Since YZ = 2(SZ),
YZ = 7 √
3.
2
Ç
9
Èäƒ
:
3
Use the formula to find the area of the triangle.
1
A=_
bh
8
2
1
=_
(
≈
63.7
2
(7 √
3)
)(
10.5
)
2
Èäƒ
9
3 ΰx Î
The area of the shaded region is
153.9 - 63.7 or approximately
90.2
square
centimeters to the nearest tenth.
HOMEWORK
ASSIGNMENT
Page(s):
Exercises:
282
Glencoe Geometry
Check Your Progress
Find the area of
the shaded region. Assume that the triangle
is equilateral. Round to the nearest tenth.
46.0 in.2
xʈ˜°
:
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Next, find the height of the triangle,
XS. Since m∠XZY is 60,
( √3
) or 10.5.
XS = 3.5 √3