Triangle Decomposition 2

Materials

Interior Angles in a Polygon worksheet, page 2 (one per person)

Straight edge (one per person)

Making Triangles Meet in Middle

Another way to find the sum of interior angle measures of a polygon is to divide it into triangles that all meet at a common vertex in the interior of the polygon. Since the common vertex is not on the polygon but in the interior, now the number of triangles is only limited by the number of sides. Below we decompose each polygon into triangles using the common vertex indicated in red. The first example on the worksheet is for demonstration purposes only since decomposing a triangle into three triangles is clearly silly given that we can only use this strategy if we already know the sum of interior angle measures in a triangle is 180 degrees.

Pattern: Sum of Interior Angle Measures

As we can see from the images above, the number of triangles in each figure corresponds to the number of sides in the polygon. Further, the triangles all come together at a common vertex in the interior of the polygon resulting in 360 degrees that do not contribute to the interior angle measures of the polygon. This means we can find the sum of interior angle measures of a polygon by adding 180 degrees for every triangle then subtracting the 360 degrees from all the triangles that don't contribute to the sum. This leads us to a general expressions for the sum of interior angle measures of a polygon with n sides:

If we compare this formula to the one in the previous activity, they are clearly equivalent if we distribute 180 degrees over two less than the number of sides in the figure. In the first decomposition strategy the two triangles in the decomposition that don't correspond to two sides by using a common vertex on the polygon total 360 degrees just as the angles of the triangles meeting at a common vertex in the interior total 360 degrees that do not contribute to the sum of interior angle measures.

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