Regular Polygons

In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. Polygons can be regular as in the Table 1 below, List of regular polygons or irregular. For a typical polygon with n sides, the sum of angles = (n – 2) * 180. This rule is derived from the fact that, any regular polygon can be divided into (n-2) triangles by drawing diagonals from one corner to all other corners. No two diagonals should cross, as in the diagram shown in Fig 1. The size of each internal angle is therefore = [(n – 2) * 180] /n.

Triangles in polygon

In a typical polygon such as a hexagon where the:
Total number of sides, n = 6
Total max number of triangles = 4 as in Fig 1. Total number of triangles 4 is given by (n – 2).
The sum of angles in a triangles = 180 °.
Therefore, the sum of angles in a polygon = (n - 2) * 180.

Typical Hexagon

The area for any regular polygon can be given in terms of the side s, the inscribed circle radius r, also known as apothem or the circumcircle radius R.
For a typical regular polygon such as the hexagon in the Fig 2, s, r and R form two triangles ABC and AEC. Using these two triangles in combination with trigonometric identities, and the internal angle θ, the area of a regular polygon can be found. α is the external angle to the polygon, it is also supplementary to θ.
The following is a step by step proof of the area of a regular polygon in terms of s, n and θ.

Extracted Triangle

The following are the other formulae for the area of the regular polygon.


Regular Polygon Area Proof

Two other formulae can be worked out in terms of r or R using the same triangle AEC from Fig 3. These formulae are shown to the left of the above table.
The centroid of a regular polygon is where all the diagonals meet. For irregular polygon, visit the Irregular Polygons page.

List of regular polygons

The table below lists some of the most popular regular polygons and some of their properties. To find out more about any of the polygons, and how to construct them and an in-depth details of their geometric properties, click on the name of the polygon. For the list of all polygons' names, visit this link List of Polygons. For other geometric shapes visit the Geometric Properties page.

Table 1

List of regular polygons (
Geomentic Shape Sum of Inerior Angles Interior
Number of
Name Shape Sides / Angles
triangle 3 180 ° 60 ° 120 ° 1
Quadrilateral 4 360 ° 90 ° 90 ° 2
Pentagon Pentagon 5 540 ° 108 ° 72 °
Hexagon Hexagon   6 720 ° 120 ° 60 ° 4
Heptagon Heptagon   7 900 ° 128.57 ° 51.43 ° 5
Octagon Octagon   8 1080 ° 135 ° 45 ° 6
Regular Polygon   n (n - 2) * 180 (n - 2) * 180/n 180-[(n-2)*180/n] (n - 2)