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Checkpoint 10
Problem 10-156
Finding Angles in and Areas of Regular Polygons
Answers to problem 10-156: a. 162º; b. 16 sides; c. ≈ 120.8 cm2
The sum of the measures of a polygon with n sides is (n – 2)180º and therefore each angle in a
regular polygon with n sides measures (n!2)180º
.
n
The sum of the exterior angles of any polygon is 360º.
To find the area of a regular polygon, use the angles and side length of one the identical triangles
that make up the polygon to find the area of the triangle. Then multiply by the number of
identical triangles to determine the area of the polygon.
Example 1: What is the measure of each interior angle of a regular 10-gon?
Solution: Using n = 10 in the formula given above
(n!2)180º
n
=
8"180º
10
= 144º
Example 2: If the regular 10-gon in Example 1 has a side length of 6 inches, what is the area of
the polygon?
Solution: The regular 10-gon is made is made up of
10 identical isosceles triangles like the
one at right. From Example 1, each
interior angle is 144º. The base angle in
the triangle is half of the interior angle so
m∠1 = 72º. We can use the tangent ratio
to find the height of the triangle:
tan 72º = h3 ! h = 3 tan 72º " 9.23 . The
area of the triangle is 12 ! 6 ! 9.23 " 27.69 .
h
1
3
3
Therefore the area of the 10-gon is 10(27.69) ! 276.9 in.2 .
Now we can go back and solve the original problems.
a.
b.
(n!2)180º
n
= 18"180º
20 = 162º
Method 1 (using the formula): 157.5º =
(n!2)180º
n
" 157.5º n = (n ! 2)180º
! 157.5º n = 180º n " 360º ! "22.5º n = "360º ! n = 16
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Method 2 (using the exterior angle): If the interior angle is 157.5º then the
exterior angle is 180º-157.5º = 22.5º. Since the sum of the exterior angles of a
polygon is 360º, 360º ÷ 22.5º = 16 sides.
c.
The octagon is made up of 8 identical triangles. Each interior angle of the
polygon measures 6!180º
8 = 135º . Using a diagram as in example 2 above,
!
135
!
m!1 = 2 = 67.5 . The base of the small triangle is 2.5 cm. To find the
h ! h = 2.5 tan 67.5º " 6.04 . The area of the large triangle
height, tan 67.5º = 2.5
is 12 ! 5 ! 6.04 " 15.1 . Eight of these triangles make up the octagon so the area of
the octagon is 8 !15.1 " 120.8 cm 2 .
Here are some more to try. Find the measure of the angle of each regular polygon.
1.
Interior angle, 12 sides
2.
Interior angle, 15 sides
3.
Interior angle, 7 sides
4.
Interior angle, 60 sides
5.
Exterior angle, 10 sides
6.
Exterior angle, 20 sides
Answer each of the following questions.
50
7.
What is the measure of each interior angle of a regular polygon with 16 sides?
8.
What is the measure of each exterior angle of a regular polygon with 16 sides?
9.
What is the area of a regular polygon with 16 sides and side length 4 inches?
10.
Each interior angle of a regular polygon measures 156! . How many sides does
it have?
11.
What is the area of a regular pentagon with side length 10 feet?
12.
Each exterior angle of a regular polygon measures 15º. How many sides does it
have?
13.
Each interior angle of a regular polygon measures 165.6º. How many sides
does it have?
14.
What is the area of a regular octagon with side length 1 meter?
15.
Each exterior angle of a regular polygon measures 13 13 º . How many sides does
it have?
16.
What is the area of a regular polygon with 15 sides and a side length of
4 inches?
Core Connections Geometry
Answers:
1.
150º
2.
156º
3.
128 47 º
4.
174º
5.
36º
6.
18º
7.
157.5º
8.
22.5º
9.
≈ 321.7 in.2
10.
15 sides
11.
≈ 344.1 ft2
12.
24 sides
13.
25 sides
14.
≈ 4.83 m2
15.
27 sides
16.
≈ 282.3 in.2
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