Interior Angles in Convex
Polygons
Bill Zahner
Lori Jordan
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Printed: May 14, 2016
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C HAPTER
Chapter 1. Interior Angles in Convex Polygons
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Interior Angles in Convex
Polygons
Here you’ll learn how to find the sum of the interior angles of a polygon and the measure of one interior angle of a
regular polygon.
Below is a picture of Devils Postpile, near Mammoth Lakes, California. These posts are cooled lava (called columnar
basalt) and as the lava pools and cools, it ideally would form regular hexagonal columns. However, variations in
cooling caused some columns to either not be perfect or hexagonal.
First, define regular in your own words. Then, what is the sum of the angles in a regular hexagon? What would each
angle be? After completing this Concept you’ll be able to answer questions like these.
Watch This
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URL: https://www.ck12.org/flx/render/embeddedobject/136939
CK-12 Foundation: Chapter6InteriorAnglesinConvexPolygonsA
Watch the first half of this video.
MEDIA
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URL: https://www.ck12.org/flx/render/embeddedobject/1267
James Sousa: Angles of Convex Polygons
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Guidance
Recall that interior angles are the angles inside a closed figure with straight sides. As you can see in the images
below, a polygon has the same number of interior angles as it does sides.
A diagonal connects two non-adjacent vertices of a convex polygon. Also, recall that the sum of the angles in a
triangle is 180◦ . What about other polygons?
Investigation: Polygon Sum Formula
Tools Needed: paper, pencil, ruler, colored pencils (optional)
1. Draw a quadrilateral, pentagon, and hexagon.
2. Cut each polygon into triangles by drawing all the diagonals from one vertex. Count the number of triangles.
Make sure none of the triangles overlap.
3. Make a table with the information below.
TABLE 1.1:
Name of Polygon
Number of Sides
Quadrilateral
Pentagon
Hexagon
4
5
6
Number of 4s from
one vertex
2
3
4
(Column 3) × (◦ in
a 4)
2 × 180◦
3 × 180◦
4 × 180◦
Total Number of
Degrees
360◦
540◦
720◦
4. Do you see a pattern? Notice that the total number of degrees goes up by 180◦ . So, if the number sides is n, then
the number of triangles from one vertex is n − 2. Therefore, the formula would be (n − 2) × 180◦ .
Polygon Sum Formula: For any n−gon, the sum of the interior angles is (n − 2) × 180◦ .
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Chapter 1. Interior Angles in Convex Polygons
A regular polygon is a polygon where all sides are congruent and all interior angles are congruent.
Regular Polygon Formula: For any equiangular n−gon, the measure of each angle is
(n−2)×180◦
.
n
Example A
Find the sum of the interior angles of an octagon.
Use the Polygon Sum Formula and set n = 8.
(8 − 2) × 180◦ = 6 × 180◦ = 1080◦
Example B
The sum of the interior angles of a polygon is 1980◦ . How many sides does this polygon have?
Use the Polygon Sum Formula and solve for n.
(n − 2) × 180◦ = 1980◦
180◦ n − 360◦ = 1980◦
180◦ n = 2340◦
n = 13
The polygon has 13 sides.
Example C
How many degrees does each angle in an equiangular nonagon have?
First we need to find the sum of the interior angles in a nonagon, set n = 9.
(9 − 2) × 180◦ = 7 × 180◦ = 1260◦
Second, because the nonagon is equiangular, every angle is equal. Dividing 1260◦ by 9 we get each angle is 140◦ .
Watch this video for help with the Examples above.
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CK-12 Foundation: Chapter6InteriorAnglesinConvexPolygonsB
Concept Problem Revisited
A regular polygon has congruent sides and angles. A regular hexagon has (6 − 2)180◦ = 4 · 180◦ = 720◦ total
degrees. Each angle would be 720◦ divided by 6 or 120◦ .
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Guided Practice
1. Find the measure of x.
2. The interior angles of a pentagon are x◦ , x◦ , 2x◦ , 2x◦ , and 2x◦ . What is x?
3. What is the sum of the interior angles in a 100-gon?
Answers:
1. From the Polygon Sum Formula we know that a quadrilateral has interior angles that sum to (4−2)×180◦ = 360◦ .
Write an equation and solve for x.
89◦ + (5x − 8)◦ + (3x + 4)◦ + 51◦ = 360◦
8x = 224
x = 28
2. From the Polygon Sum Formula we know that a pentagon has interior angles that sum to (5 − 2) × 180◦ = 540◦ .
Write an equation and solve for x.
x◦ + x◦ + 2x◦ + 2x◦ + 2x◦ = 540◦
8x = 540
x = 67.5
3. Use the Polygon Sum Formula. (100 − 2) × 180◦ = 17, 640◦ .
Interactive Practice
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Explore More
1. Fill in the table.
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Chapter 1. Interior Angles in Convex Polygons
TABLE 1.2:
# of sides
3
4
5
6
7
8
9
10
11
12
2.
3.
4.
5.
6.
7.
8.
9.
Sum of the Interior Angles
360◦
540◦
Measure of Each Interior Angle in a Regular
n−gon
60◦
108◦
120◦
What is the sum of the angles in a 15-gon?
What is the sum of the angles in a 23-gon?
The sum of the interior angles of a polygon is 4320◦ . How many sides does the polygon have?
The sum of the interior angles of a polygon is 3240◦ . How many sides does the polygon have?
What is the measure of each angle in a regular 16-gon?
What is the measure of each angle in an equiangular 24-gon?
Each interior angle in a regular polygon is 156◦ . How many sides does it have?
Each interior angle in an equiangular polygon is 90◦ . How many sides does it have?
For questions 10-18, find the value of the missing variable(s).
10.
11.
12.
13.
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14.
15.
16.
17.
18.
19. The interior angles of a hexagon are x◦ , (x + 1)◦ , (x + 2)◦ , (x + 3)◦ , (x + 4)◦ , and (x + 5)◦ . What is x?
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 6.1.
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