Triangles of a Polygon

Definition: The triangles of a polygon are the triangles created by drawing line segments from one vertex of a polygon to all the others.
Try this Adjust the number of sides of the polygon below, or drag a vertex to note the number of triangles inside the polygon.

Regular Polygon case

In the case of regular polygons, the formula for the number of triangles in a polygon is: where
n  is the number of sides (or vertices)

Why? The triangles are created by drawing the diagonals from one vertex to all the others. Since there would be no diagonal drawn back to itself, and the diagonals to each adjacent vertex would lie on top of the adjacent sides, the number of diagonals from a single vertex is three less the the number of sides, or n-3. The number of triangles is one more than that, so n-2.

This can be used as another way to calculate the sum of the interior angles of a polygon. The interior angles of a triangle always sum to 180°. The number of triangles is n-2 (above). Therefor the interior angles of the polygon must be the sum of all the triangles' interior angles, or 180(n-2).

Irregular Polygon case

For convex , irregular polygons, dividing it into triangles can help if you trying to find its area. For example, in the figure on the right, it may be possible to find the area of each triangle and then sum them.

For most concave , irregular polygons , the triangles are of little practical use.