Polygon Formula

Before starting with the polygon formula, let us recall the definition of a polygon. A polygon is a closed 2-D shape that has three or more straight lines. A polygon should have at least three sides. Each side of the line segment intersects with another line segment at the vertex. Let us learn more about the different polygons and their formulas. 

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 different values
• Convex polygon – All the interior angles of a polygon < 180 degrees
• Concave Polygon – Polygons that have one or more interior angles with a measure of >180 degrees

The number of sides of a polygon determines its shape and it's named after its number of sides. Common examples of polygons are triangles, squares, pentagons, hexagons, etc. Below are the listed polygons based on their number of sides. 

What is Polygon Formula?

The important formulas associated with a regular polygon are given below: 

Formula 1: For a regular 'n' sided polygon, the sum of interior angles of a polygon is 180°(n-2)

Formula 2: The number of diagonals of an “n-sided” polygon = [n(n-3)]/2

Formula 3: The measure of each interior angle of a regular n-sided polygon = [(n-2)180°]/n

Formula 4: The measure of exterior angles of a regular n-sided polygon  = 360°/n 

Formula 5: Area of regular polygon = (number of sides × length of one side × apothem)/2, where, the length of apothem is given as the \(\dfrac{l}{2\tan(\dfrac{180}{n})}\) and where l is the side length and n is the number of sides of the regular polygon.

Formula 6: In terms of the perimeter of a regular polygon, the area of a regular polygon is given as, Area = (Perimeter × apothem)/2, in which perimeter = number of sides × length of one side

Properties of Polygon

The important properties of the polygon are

• The sum of the interior angles of all the quadrangles = 360°.
• If at least one of the interior angles is > 180 degrees, then it is called a concave polygon.
• 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.

Examples Using Polygon Formulas

Example 1: Find the sum of the interior angle of a hexagon.

Solution:

We know that a hexagon has six sides.

Using the polygon formula, we know that the sum of interior angles is given by:

Interior angle sum = 180°(n-2)

= 180°(6-2)

= 180° (4)

= 720°

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 the perimeter and value of one interior angle.

Solution:

The polygon is an octagon. Hence, n = 8

Length of each side, s = 7 cm

The perimeter of the octagon is P = n × s

P = 8 × 7

   = 56 cm

Now, to find each interior angle by using the polygon formula,

Interior Angle = [(n-2)180°]/n

= [(8 - 2)180°] / 8

= (6 × 180°) / 8

= 135°

Thus, the perimeter of the given octagon is 56 cm and the value of each internal angle is 135 degrees.

Example 3: Using the polygon formula, find the sum of the interior angle of a triangle.

Solution: 

We know that a triangle has three sides.

Using the polygon formula, we know that the sum of interior angles is given by:

Interior angle sum = 180°(n-2)

= 180°(3-2)

= 180° (1)

= 180°

Hence, the sum of the interior angles of a triangle is 180°.