Triangle Decomposition 1

Materials

Interior Angles in a Polygon worksheet, page 1 (one per person)

Straight edge (one per person)

Making Triangles from a Vertex

One way to find the sum of interior angle measures of a polygon is to divide it into triangles. This can be done many different ways for different polygons (especially as the number of sides increases). If we are trying to assess a pattern in the sum of interior angle measures as the number of sides increases, it helps if we do so in a systematic manner. This activity suggests that each of our triangles share a common vertex. The number of triangles is limited by how many vertices are not adjacent to the common vertex because lines connecting adjacent vertices will not form a triangle. Below we decompose each polygon into triangles using the common vertex indicated in red using, as a starting point, that the sum of interior angle measures of a triangle is 180 degrees.

Pattern: Sum of Interior Angle Measures

As we can see from the images above, a pattern emerges regarding the number of triangles in relation to the number of sides on the polygon. Each polygon after the triangle can be decomposed into a number of triangles that is two fewer than the number of sides in the figure. This leads us to a general expressions for the sum of interior angle measures of a polygon with n sides:

Note on Convex VS Concave Polygons

All of the examples we are exploring in this activity are convex (no angles are greater than 180 degrees). The strategies for finding the sum of interior measures of a polygon we will use in these activities do not necessarily work for concave (not convex) polygons. In the case of the first two activities, it is because not every straight line is contained in the interior of the shape. In the last activity, we need a broader understanding of exterior angles. However, as it turns out, the pattern we discover still holds for concave polygons, primarily because every concave polygon can be decomposed into triangles (or even just convex polygons), just not in the same systematic manner.

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