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Find that interior angle measure!
Name: __________________________
In this activity, we are going to explore interior angles in polygons. We will be exploring the relationship
between the sum of a polygon’s interior angles and the number of sides it has.
Part 1: Exploring the properties of the angles of a triangle.
First, we will first explore some of the properties of a polygon that we should know very well by now –
the triangle. Let’s first review some of those properties.
1. What is the sum of the angles of a triangle? Does it matter what kind of triangle it is?
Now let’s get into the GeoGebra program. Open a new window. Using the polygon tool
, make a
three-sided figure (also called a triangle!) by connecting three separate points in a clockwise manner
(this is important!).
Now, to measure all the angles of the triangle, use the angle tool
and just click on the center of the
triangle to get the measure of all three angles. You should have something like this:
What is the sum of the measures of the angles? To find out, let’s use the input tool down on the bottom
of the screen. We want to add all of the angles together. If you notice, each angle of the triangle has a
separate variable. Using those variables, we can find their sum by pressing enter after adding up all the
variables. (Variables can be found in the middle drop down menu to the right of the bar)
The sum will come out as one of the “dependent objects” on the left side of the screen, under some
variable. Try moving the points around and you will be able to confirm your answer from question 1.
Part 2: What about other polygons’ sum of their interior angles?
Now, we know what the sum of the interior angles of a triangle is. But what about the sum of the
interior angles of more sided figures, such as the rectangle or the hexagon? Furthermore, can we find
the individual interior angles of a regular polygon?
In case we forgot, a regular polygon is a polygon where all the angles are the same measure and all the
sides are the same length.
Opening up a new page, let’s build a four-sided polygon. However, this time, let’s build a regular
polygon. On the bottom corner of the polygon tool, click on the small arrow pointing downwards.
Select regular polygon
. To use this tool, all you have to do is create any two points and a window
will pop up asking for the number of sides you would like your polygon to have. In our case, we want 4
sides to get a four-sided polygon.
Now as before, let’s measure the sum of the angles by using the input bar (otherwise you can use a
calculator). Also, note what the measures of the interior angles are. Now that we have done so for the 4sided polygon, let’s do the same thing for larger sided polygons. Fill out the table on the next page.
Polygon
(Regular)
Number of Sides
Sum of the interior
angles
Measure of a single
interior angle
Equilateral Triangle
Square
Pentagon
Hexagon
2. Without using GeoGebra, what do you think the sum of the interior angles in a regular octagon
would be? ________ What would the measure of one of its single angles be? ______ Explain.
Part 3: Triangles in Polygons?
In this last section, we will try to make a formula for finding the sum of interior angles, regardless of how
many sides the polygon has. Close the program since we are now done with it. Now, you may have
noticed something about the sum of the interior angles as the polygon increases in size.
3. Describe the pattern you see in the interior angle sum as we increase the number of sides.
4. What do you think is the reason for this pattern?
On the next page, there are a couple of polygons in order of increasing number of sides. We are going to
be counting the number of triangles there are in each of the polygons. To do so, choose one vertex on a
polygon. From that chosen vertex, draw a line to all the other vertices of the polygon. Then, count the
number of triangles in each of the polygon made your lines. Fill in the chart below.
Polygon
Number of Sides
Number of Triangles Inside
Triangle
3
1
Square
Pentagon
Hexagon
Interior Angle Sum
(Copy from last table)
Part 4: Putting it all together
5. What is the relationship between the number of sides of a polygon and the number of triangles
inside the same polygon? _________________ Complete this sentence: If there are N sides,
then there are __________ triangles formed by diagonals from one vertex.
We will now try to find the formula for determining the sum of the interior angles to the number of
sides.
6. Using your answer from question #5, try and complete this sentence: If a polygon has N sides,
then the sum of its interior angles is _____________ (Hint: Look at your last table. Can you find
a relationship between the number of triangles and the interior angle sum?)
7. Using your formula, what would the interior angle sum be for an octagon? _________ Does this
match your answer from question #2?
8. Complete the chart below to summarize the formulas you have been discovering.
Number of sides
N-gon
Number of triangles
Sum of the interior
angles
Measure of a single
interior angle
N
Part 5: Applying! Show work for each problem 
9. In a regular 76-sided polygon, the measure of one of its interior angles is ___________.
10. The angle sum of a polygon is 6120 degrees , how many sides does it have? _________.
11. The sum of the interior angles of 1 100-gon would be ______________.