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Title
Sum of Interior Angles of Polygons
Problem Statement
What is the sum of interior angles of a triangle? Of a quadrilateral? Of a pentagon? Of a hexagon? What is the sum of interior angles of convex polygons in terms of the number of sides?
Problem setup

Plans to Solve/Investigate the Problem
Using Geometer’s Sketchpad, I will construct each shape and calculate the sums of interior angles of each of them. I will also construct the diagonals of each polygon to see if that will help.

Investigation/Exploration of the Problem
The first step is to accurately construct each shape. This may be done by using software, such as Geometer’s Sketchpad, or a compass, straightedge, and protractor on paper. Once the polygons have been constructed correctly we can begin to calculate the interior angles. Start with the triangle. After calculating the interior angles, find their sum. Notice that the triangle has no diagonals to construct. Now that the sum of interior angles of a triangle is known, we may use this information as a tool to find the interior angles of any convex polygon.

To calculate the interior angles of any convex polygon, start by constructing all the diagonals leaving from any single vertex. That is all the diagonals leaving from one vertex only. In doing this, notice that in each shape a certain number of triangles are formed in the interior by the diagonals; two in the quadrilateral, three in the pentagon, and four in the hexagon. Notice that the number of triangles formed by the diagonals is always two less than the number of sides. Since the sum of interior angles of a triangle is known, we may simply multiply by the number of triangles formed in the interior by the diagonals to find the sum of interior angles of the entire shape.

For the quadrilateral, the sum of interior angles would be: 2; the number of triangles formed in the interior by the diagonals. Multiplied by 180 degrees; the sum of interior angles of a triangle. Working out the problem would give you 360 degrees. The sum of interior angles of the pentagon would be 540 degrees; the number of triangles formed in the interior by the diagonals: 3, multiplied by the sum of interior angles of a triangle: 180 degrees. For the hexagon, the sum of interior angles would be: 4; the number of triangles formed in the interior by the diagonals. Multiplied by 180 degrees; the sum of interior angles of a triangle. Working out the problem would give you 720 degrees.

Extensions of the Problem
Does the relationship for convex polygons also hold true for non-convex polygons? Explain.

Author & Contact
Nathan Little
nlittle@aug.edu

Link(s) to resources, references, lesson plans, and/or other materials
My constructions
Intermath

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