Geometry: Polygons
Properties of Polygons
Diagonals
One property of all convex polygons has to do with the number of
diagonals that it has:
Every convex polygon with n sides has n(n-3)/2 diagonals.
With this formula, if you are given either the number of diagonals or the number
of sides, you can figure out the unknown quantity. Diagonals become useful in
geometric proofs when you may need to draw in extra
lines or
segments, such as diagonals.
Interior Angles
The interior angles of polygons follow certain patterns based on the number of sides, too. First of all, a polygon with n sides has n vertices, and therefore has n interior angles. The sum of these interior angles is equal to 180(n-2) degrees. Knowing this, given all the interior angle measures but one, you can always figure out the measure of the unknown angle.
Exterior Angles
An exterior angle on a polygon is formed by extending one of the sides of the polygon outside of the polygon, thus creating an angle supplementary to the interior angle at that vertex. Because of the congruence of vertical angles, it doesn't matter which side is extended; the exterior angle will be the same.
The sum of the exterior angles of any polygon (remember only convex polygons are being discussed here) is 360 degrees. This is a result of the interior angles summing to 180(n-2) degrees and each exterior angle being, by definition, supplementary to its interior angle. Take, for example, a triangle with three vertices of 50 degrees, 70 degrees, and 60 degrees. The interior angles sum to 180 degrees, which equals 180(3-2). Because the exterior angles are supplementary to the interior angles, they measure, 130, 110, and 120 degrees, respectively. Summed, the exterior angles equal 360 degreEs.
A special rule exists for regular polygons: because they are equiangular, the exterior angles are also congruent, so the measure of any given exterior angle is 360/n degrees. As a result, the interior angles of a regular polygon are all equal to 180 degrees minus the measure of the exterior angle(s).
Notice that the definition of an exterior angle of a polygon differs from that of an exterior angle in a plane. A polygon's exterior angle is not equal to 360 degrees minus the measure of the interior angle. A polygon's interior and exterior angles at a given vertex don't span the entire plane, they only span half the plane. That is why they are supplementary--because their measures sum to 180 degrees instead of 360.
