6.1: Interior Angles in Convex Polygons
Below is a picture of Devil’s Post pile, 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 pentagonal.
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
CK12 Foundation: Chapter6InteriorAnglesinConvexPolygonsA
Watch the first half of this video.
James Sousa: Angles of Convex Polygons
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 nonadjacent vertices of a convex polygon. Also, recall that the sum of the angles in a triangle is
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.
Name of Polygon  Number of Sides 
Number of 
(Column 3) 
Total Number of Degrees 

Quadrilateral  4  2 


Pentagon  5  3 


Hexagon  6  4 


4. Do you see a pattern? Notice that the total number of degrees goes up by
Polygon Sum Formula: For any
A regular polygon is a polygon where all sides are congruent and all interior angles are congruent.
Regular Polygon Formula: For any equiangular
Example A
Find the sum of the interior angles of an octagon.
Use the Polygon Sum Formula and set
Example B
The sum of the interior angles of a polygon is
Use the Polygon Sum Formula and solve for
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
Second, because the nonagon is equiangular, every angle is equal. Dividing
Watch this video for help with the Examples above.
CK12 Foundation: Chapter6InteriorAnglesinConvexPolygonsB
Concept Problem Revisited
A regular polygon has congruent sides and angles. A regular hexagon has
Vocabulary
The interior angle of a polygon is one of the angles on the inside. A regular polygon is a polygon that is equilateral (has all congruent sides) and equiangular (has all congruent angles).
Guided Practice
1. Find the measure of
2. The interior angles of a pentagon are
3. What is the sum of the interior angles in a 100gon?
Answers:
1. From the Polygon Sum Formula we know that a quadrilateral has interior angles that sum to
Write an equation and solve for
2. From the Polygon Sum Formula we know that a pentagon has interior angles that sum to
Write an equation and solve for
3. Use the Polygon Sum Formula.
Practice
 Fill in the table.
# of sides  Sum of the Interior Angles 
Measure of Each Interior Angle in a Regular 


3 


4 


5 



6 


7  
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 \begin{align*}156^\circ\end{align*}. How many sides does it have?
 Each interior angle in an equiangular polygon is \begin{align*}90^\circ\end{align*}. How many sides does it have?
For questions 1018, find the value of the missing variable(s).
 The interior angles of a hexagon are \begin{align*}x^\circ, (x + 1)^\circ, (x + 2)^\circ, (x + 3)^\circ, (x + 4)^\circ,\end{align*} and \begin{align*}(x + 5)^\circ.\end{align*} What is \begin{align*}x\end{align*}?
Image Attributions
Description
Learning Objectives
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.