A polygon by definition is any geometric shape that is enclosed by a number of straight sides, and a polygon is considered regular if each side is equal in length. Polygons are classified by their number of sides. For example, a six-sided polygon is a hexagon, and a three-sided one is a triangle.
The number of sides of a regular polygon can be calculated by using the interior and exterior angles, which are, respectively, the inside and outside angles created by the connecting sides of the polygon. For a regular polygon the measure of each interior angle and each exterior angle is congruent. For example, a regular octagon has interior angles each equal to 125 degrees.
These relationships only hold true for convex polygons where the measure of each interior angle does not exceed 180 degrees.
Using Interior Angles
Subtract the interior angle from 180; then divide 360 by the difference of the angle and 180 degrees. For example, if the interior angle was 165, subtracting it from 180 would yield 15, and 360 divided by 15 equals 24, which is the number of sides of the polygon. Here is the general formula (it is important to note that this only works for the interior angles of a regular polygon):
Using Exterior Angles
Divide 360 by the amount of the exterior angle to also find the number of sides of the polygon. For example, if the measurement of the exterior angle is 60 degrees, then dividing 360 by 60 yields 6. Six is the number of sides that the polygon has. This is a hexagon, so we can check this reasoning by finding the interior angle to be 120 degrees, which is the measure of the interior angle of a hexagon.
The general formula using the exterior angles of a regular polygon follows:
Subtracting the interior angle from 180 gives the exterior angle, and subtracting the exterior angle from 180 gives the interior angle because these angles are adjacent.
Not all polygons have congruent angles and sides. The measure of the internal angles can vary depending on the measures of each side. Regardless of the polygon shape, the sum of exterior angles will always be 360 degrees. We can use this relationship to reason out a formula for an n-sided polygon with any side lengths.
The sum of the interior angles of a polygon can be related to the the number of sides through the polygon formula:
We can try this formula with any quadrilateral. We know that the sum of the interior angles of any four sided polygon (like a square, rhombus, parallelogram, or trapezoid) is 360 degrees. Plugging this into the formula we can prove this known relationship:
This formula for any polygon works for either a convex or concave polygon.
Terminology of Polygons
As a helpful guide for reporting calculations, these are the general conventions for discussing polygons in geometry and trigonometry.
- Line segments make up each side of a polygon. They are straight lines of determined length.
- An apothem is a straight line from the center of a regular polygon to any side that forms a right angle with that side.
Naming polygons (3 - 10 sides):
- 3 sides – triangle
- 4 sides – square
- 5 sides – pentagon
- 6 sides – hexagon
- 7 sides – heptagon
- 8 sides – octagon
- 9 sides – nonagon
- 10 sides – decagon
About the Author
Chance E. Gartneer began writing professionally in 2008 working in conjunction with FEMA. He has the unofficial record for the most undergraduate hours at the University of Texas at Austin. When not working on his children's book masterpiece, he writes educational pieces focusing on early mathematics and ESL topics.