Polygon Formula
Before starting with the polygon formula, let us recall the definition of a polygon. A polygon is a closed 2D figure which comprises three or more straight lines. A polygon should have at least three sides. Each side of the line segment must intersect with another line segment only at its endpoint. Based on the number of sides of a polygon, we can easily identify the polygon shape.Common examples of polygons are triangles, squares, pentagons, hexagons, etc.
Types of Polygon
Based on the angle measure and the sides of a polygon, the polygon is classified into:
 Regular Polygon – All the interior angles and the sides are of the same measure
 Irregular Polygon – All the interior angles and the sides have diferent values
 Convex polygon – All the interior angles of a polygon are strictly less than 180 degrees
 Concave Polygon – Polygons that have one or more interior angles with a measure of more than 180 degrees
Below is the listed polygons based on their number of sides.
Let us now understand the polygon formula using a few solved examples in the following section.
What is Polygon Formula?
The important formulas associated with a polygon are given below:
Formula 1:
For a regular 'n' sided polygon, the sum of interior angles of a polygon can be calculated using formula,
The sum of interior angles of a polygon with “n” sides = 180°(n2)
Formula 2:
Number of diagonals of a “nsided” polygon can be calculated using the formula,
Number of diagonals of a “nsided” polygon = [n(n3)]/2
Formula 3:
The measure of each interior angle of a regular nsided polygon can be calculated using the formula,
The measure of each interior angle of a regular nsided polygon = [(n2)180°]/n
Formula 4:
The measure of exterior angles of a regular nsided polygon can be calculated using the formula,
The measure of exterior angles of a regular nsided polygon = 360°/n
Properties of Polygon
The important properties of the polygon are
 The sum of the interior angles of all the quadrangles is equal to 360 degrees.
 If at least one of the interior angles is greater than 180 degrees, then it is called concave
 If a polygon does not cross over itself and has only one boundary, it is called a simple polygon. Otherwise, it is a complex polygon
Solved Examples Using Polygon Formula

Example 1:
Find the sum of the interior angle of a hexagon.
Solution:
We know that a hexagon has six sides.
The formula to find the sum of interior angles is given by:
Interior angle sum = 180°(n2)
= 180°(62)
= 180° (4)
= 720°
Answer: Hence, the sum of the interior angles of a hexagon is 720°.

Example 2:
A polygon is an octagon and its side length is 7 cm. Calculate its perimeter and value of one interior angle.
Solution:
The polygon is an octagon. Hence, n = 8
Length of one side,
s = 7 cm
The perimeter of the octagon
P = n × s
P = 8 × 7
= 56 cm
Now, to find each interior angle,
Interior Angle = [(n2)180°]/n
= [(8  2)180°] / 8
= (6 × 180°) / 8
= 135°
Answer: Thus, the perimeter of the given octagon is 56 cm and the value of each internal angle is 135 degrees.