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Lesson 10-3
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 10-6: Area of a Regular Polygon
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The area of a regular polygon is
a
p
A
The center of a regular polygon is
The radius of a regular polygon is
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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 m2 360
Divide 360 by the number of sides.
The apothem bisects the vertex angle of
m1
the
1
2
3
triangle formed by
the radii.
m2 (
)
m3 180 (
m1 , m2 188
Geometry Lesson 10-3
Substitute
)
for m1.
The sum of the measures of the angles of a triangle is
.
, and m3 Daily Notetaking Guide
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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.
23.5 ft
Step 1 Find the apothem a.
a2 (
)2 (
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a2 )2
9.0 ft
9.0 ft
Pythagorean Theorem
Solve for a.
a2 a<
Step 2 Find the perimeter p.
Find the perimeter, where n the number of sides
of a regular polygon.
p ns
(
p (8)
)
Substitute
for n and
for s, and simplify.
Step 3 Find the area A.
A 12 ap
A 12
(
Area of a
)(
A
Substitute
for a and
for p.
Simplify.
To the nearest 10 ft 2, the area is
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)
polygon
ft 2.
3 Applying Theorem 10-6 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
( )(
A 5 12 ap
A 5 12
( )(
for a.
Multiply each side by
.
Find the perimeter.
)
Substitute
for n and
for s, and simplify.
Area of a regular polygon
)
A5
The area of the equilateral triangle is
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s
? shorter leg
Substitute
for a and
for p.
Simplify.
cm 2.
Geometry Lesson 10-3
189
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Quick Check.
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
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3
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2. Find the area of a regular pentagon with 11.6-cm sides and an 8-cm
apothem.
3. The side of a regular hexagon is 16 ft. Find the area of the hexagon.
190
Geometry Lesson 10-3
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