InterMath | Workshop Support | Write Up Template
Title
Triangles in a Polygon
Problem Statement
Recommended Investigations | Sum of Angles in a Polygon
What is the sum of the angles of a triangle? of a quadrilateral? of a pentagon? of a
hexagon? What is the sum of the angles in convex polygons in terms of the number of
sides? Move the mouse pointer over the picture for a hint.
Problem setup
How can you find the sums of the interior angles of a polygon? Make a comparison between the
numbers of sides a polygon with the sum of the angles. You have to use a convex polygon.
Plans to Solve/Investigate the Problem
You will need to construct a convex polygon on geo-sketch pad. You can measure all the
interior angles and find the sum of the angles. You can construct triangles from a single vertex
and add the sum of the angles of the triangles. Find the relationship between the sum of the
interior angles and the sum of the angles of the triangles inside the polygon. You can use geosketch pad or excel.
Investigation/Exploration of the Problem
You have to begin by defining a convex polygon. A convex polygon is a polygon that has all of
its interior angles less than 180 degrees. Note, this is not the sum of the interior angles. You can
construct a convex polygon using GSP. After you construct the polygon, you can measure the
interior angles of the polygon. You find the sum of the interior angles of the polygon. You can
use the measure tool and calculate the sum. You will then need to find one vertex of the
polygon. You can connect the vertex with all but the two adjoining vertices. These connecting
line segments will construct triangles inside the polygon. The two adjoining vertices would only
make a flat line. By definition, we know that the sum of the interior angles of a triangle will
always be 180’. You can take the number of triangles and multiply this by 180’. This number
will match the sum of the interior angles of the polygon. See the drawing from GSP.
m ABC = 118°
m BCD = 107°
m CDE = 135°
m DEF = 150°
m EFA = 87°
m FAB = 123°
m ABC + m BCD + m CDE + m DEF + m EFA + m FAB = 720°
m EFA = 87°
The polygon is convex because none of the interior angles are equal to or greater than 180’. The
sum of the interior angles of the polygon using GSP is 720’. You need to find the number of
sides of the polygon. This polygon has 6 sides. You will see that using a 6 sided polygon, you
will have four triangles inside the polygon. Remember, the sum of the interior of a triangle is
180’. If you multiply the number of triangles by 180’ you will come up with 720’. This is the
same as the sum of the interior angles of the polygon. The number of triangles will always be 2
less than the number of sides of the polygon. Therefore, a formula can be used to find the sum of
the interior angles of a convex polygon as long as you know how many sides are in the polygon.
The formula would be (n-2)*180 with n being the number of sides. If you use excel, you could
show a relationship between the number of sides and the sum of the interior angles. This would
be a simple way to find the sum using the number of sides. See the excel chart.
# of sides # of triangles #VALUE!
6
4
720
7
5
900
8
6
1080
9
7
1260
10
8
1440
11
9
1620
12
10
1800
The first column is number of sides of the polygon. The second column is the number of
triangles in that polygon. The third column is the formula (n-2)*180. You could continue this
and find the sum of a polygon that has 30 sides or 130 sides. Remember, the polygon has to
convex.
Author & Contact
Carole C. Jackson
cjackson@rockdale.k12.ga.us
carolecjackson@yahoo.com