Question

Find the sum of the measures of the interior angles of a decagon.

Answer 1

Start with vertex A and connect it to all other vertices (it is already connected to B and E by the sides of the figure). Three triangles are formed. The sum of the angles in each triangle contains 180°. The total number of degrees in all three triangles will be 3 times 180. Consequently, the sum of the interior angles of a pentagon is:
3 180 = 540
Notice that a pentagon has 5 sides, and that 3 triangles were formed by connecting the vertices. The number of triangles formed will be 2 less than the number of sides.

This pattern is constant for all polygons. Representing the number of sides of a polygon as n, the number of triangles formed is (n - 2). Since each triangle contains 180°, the sum of the interior angles of a polygon is 180(n - 2).