<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

Interior Angles in Convex Polygons

Angles inside a closed figure with straight sides.

Atoms Practice
%
Progress
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
%
Practice Now
Turn In
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 gon, the interior angles add up to

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 gon, the measure of each angle is .

In the picture below, if all eight angles are congruent then each angle is .

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 and . What is ?

From the Polygon Sum Formula we know that a pentagon has interior angles that sum to .

Write an equation and solve for .

Example 2

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

Use the Polygon Sum Formula. .

Example 3

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

Use the Polygon Sum Formula and solve for .

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

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

Example 5

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

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

Review

  1. Fill in the table.
# of sides Sum of the Interior Angles Measure of Each Interior Angle in a Regular gon
3
4
5
6
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 . How many sides does the polygon have?
  4. The sum of the interior angles of a polygon is . 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 . How many sides does it have?
  8. Each interior angle in an equiangular polygon is . 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 and What is ?

Review (Answers)

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

Resources

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Vocabulary

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.

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Interior Angles in Convex Polygons.
Please wait...
Please wait...