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?
Interior Angles in Convex Polygons
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
Finding the Sum of Interior Angles
1. Find the sum of the interior angles of an octagon.
Use the Polygon Sum Formula and set
2. The sum of the interior angles of a polygon is
Use the Polygon Sum Formula and solve for
Watch the first half of this video.
Measuring Angles
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
Earlier Problem Revisited
A regular polygon has congruent sides and angles. A regular hexagon has
Examples
Example 1
Find the measure of
From the Polygon Sum Formula we know that a quadrilateral has interior angles that sum to
Write an equation and solve for
Example 2
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 3
What is the sum of the interior angles in a 100gon?
Use the Polygon Sum Formula.
Review
 Fill in the table.
# of sides  Sum of the Interior Angles  Measure of Each Interior Angle in a Regular 


3  
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5  
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10  
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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*}?
Review (Answers)
To view the Review answers, open this PDF file and look for section 6.1.