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Interior Angles in Convex Polygons

Angles inside a closed figure with straight sides.

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Interior Angles in Convex Polygons

Interior Angles in Convex Polygons

The interior angle of a polygon is one of the angles on the inside, as shown in the picture below. A polygon has the same number of interior angles as it does sides.

The sum of the interior angles in a polygon depends on the number of sides it has. The Polygon Sum Formula states that for any ngon, the interior angles add up to (n2)×180.

n=8(82)61×180×180080

Once you know the sum of the interior angles in a polygon it is easy to find the measure of ONE interior angle if the polygon is regular: all sides are congruent and all angles are congruent. Just divide the sum of the angles by the number of sides.

Regular Polygon Interior Angle Formula: For any equiangular ngon, the measure of each angle is (n2)×180n.

In the picture below, if all eight angles are congruent then each angle is (82)×1808=6×1808=10808=135.

What if you were given an equiangular seven-sided convex polygon? How could you determine the measure of its interior angles?

 

Examples

Example 1

The interior angles of a pentagon are x,x,2x,2x, and 2x. What is x?

From the Polygon Sum Formula we know that a pentagon has interior angles that sum to (52)×180=540.

Write an equation and solve for x.

x+x+2x+2x+2x8xx=540=540=67.5

Example 2

What is the sum of the interior angles in a 100-gon?

Use the Polygon Sum Formula. (1002)×180=17,640.

Example 3

The interior angles of a polygon add up to 1980. How many sides does it have?

Use the Polygon Sum Formula and solve for n.

(n2)×180180n360180nn=1980=1980=2340=13

The polygon has 13 sides.

Example 4

How many degrees does each angle in an equiangular nonagon have?

First we need to find the sum of the interior angles; set n=9.

(92)×180=7×180=1260

“Equiangular” tells us every angle is equal. So, each angle is 12609=140.

Example 5

An interior angle in a regular polygon is 135. How many sides does this polygon have?

Here, we will set the Regular Polygon Interior Angle Formula equal to 135 and solve for n.

(n2)×180n180n360360n=135=135n=45n=8The polygon is an octagon.

Review

  1. Fill in the table.
# of sides Sum of the Interior Angles Measure of Each Interior Angle in a Regular ngon
3 60
4 360
5 540 108
6 120
7
8
9
10
11
12
  1. What is the sum of the angles in a 15-gon?
  2. What is the sum of the angles in a 23-gon?
  3. The sum of the interior angles of a polygon is 4320. How many sides does the polygon have?
  4. The sum of the interior angles of a polygon is 3240. How many sides does the polygon have?
  5. What is the measure of each angle in a regular 16-gon?
  6. What is the measure of each angle in an equiangular 24-gon?
  7. Each interior angle in a regular polygon is 156. How many sides does it have?
  8. 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).

  1. The interior angles of a hexagon are x,(x+1),(x+2),(x+3),(x+4), and (x+5). What is x?

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.1. 

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Vocabulary

TermDefinition
Interior angles Interior angles are the angles inside a figure.
Polygon Sum Formula The Polygon Sum Formula states that for any polygon with n sides, the interior angles add up to (n-2) \times 180 degrees.

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