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
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
In the picture below, if all eight angles are congruent then each angle is
What if you were given an equiangular sevensided convex polygon? How could you determine the measure of its interior angles?
Examples
Example 1
The interior angles of a pentagon are
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 100gon?
Use the Polygon Sum Formula.
Example 3
The interior angles of a polygon add up to
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
Here, we will set the Regular Polygon Interior Angle Formula equal to
Review
 Fill in the table.
# of sides  Sum of the Interior Angles  Measure of Each Interior Angle in a Regular 


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8  
9  
10  
11  
12 
 What is the sum of the angles in a 15gon?
 What is the sum of the angles in a 23gon?
 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 16gon?
 What is the measure of each angle in an equiangular 24gon?
 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 1018, find the value of the missing variable(s).
 The interior angles of a hexagon are
x∘,(x+1)∘,(x+2)∘,(x+3)∘,(x+4)∘, and(x+5)∘. What isx ?
Review (Answers)
To see the Review answers, open this PDF file and look for section 6.1.