How to find the Area of a Regular Polygon?

Geometry is a branch of mathematics that involves the calculation of different parameters of plane shapes and solid shapes. In this article, we have discussed formulas to calculate parameters like area, perimeter and volume of different shapes and have briefly described regular polygons and formulas to calculate its area along with some sample solutions for better understanding.  

Some Basic formulas for plane shapes 

Rectangle

• Area = length × breadth
• Perimeter = 2(length+breadth)

Square

• Area = (side)²
• Perimeter = 4(side)

Circle

• Diameter = 2 × radius
• Area = π × (radius)2

Triangle

• Area = 1/2 breadth × height

Solid  shapes

Cube

• Volume = (side)³
• Lateral surface area = 4 × (side)²
• Total surface area = 6 × (side)²

Cuboid

• Volume = length × breadth × height
• Lateral surface area = 2 × height(l+b)
• Total surface area = 2(lb+lh+hb)

Sphere

• Volume = 4/3πr³
• Surface area = 4πr²

Cone

• Volume = 1/3πr²h
• Total surface area = πr (l+radius)

What is a Regular Polygon?

A regular polygon includes plane shapes having equal-length sides with equal interior angles. Rhombus, square, equilateral triangle, rectangle are some basic examples of regular polygons. Any other polygon having unequal sides or interior angles is known as irregular polygons which include scalene triangle, trapezium, etc.

Properties of a regular polygon

• A regular polygon has all sides and interior angles equal.
• The bisectors of a regular polygon meet at the centre moving from the interior angles.
• The distance between the centre and the vertices of a regular polygon are equal.
• The length of the perpendicular drawn from the centre to the vertices of a regular polygon is always equal.

Area of a Regular Polygon

All the regular polygons can be considered as a cyclic polygon and tangential polygon as all the vertices of a regular polygon lie on a circumscribed circle viz. they have concyclic points and they also have an inscribed circle that is tangent to every side lies at the midpoint respectively. as a polygon has equal sides and equal angles usually an apothem is used to calculate the area of a regular polygon.

An apothem is a line segment that joins the centre of the polygon to the midpoints of the sides and is drawn perpendicular to the sides.

The area of a regular polygon can be written as

where, 

l is the length of a side

n is the number of sides

Sample Problems

Problem 1. Calculate the area of 5 sided polygon with a side length of 4cm.

Solution:

The given parameters are,

l = 4 cm and n = 5

The formula for finding the area.

=>

=>A = (4)² × 5/4tan(180/5)

=>A = 80/4 × 0.7265

=>A = 27.53cm²

Problem 2. Calculate the area of 6 sided polygon with a side length of 10cm.

Solution:

The given parameters are

l=10cm and n=6

The formula for finding the area

=>

=>A = (10)²×6/4tan(180/6)

=>A = 600/4 × 0.5773

=>A = 259.83cm²

Problem 3. Calculate the area of 3 sided polygon with a side length of 12cm.

Solution:

The given parameters are 

l=12cm and n=3

The formula for finding the area

=>

=>A = (12)² × 3/4tan(180/3)

=>A = 432/4 × 1.7320

=>A = 62.35cm²

Problem 4. Calculate the area of 4 sided polygon with a side length of 4cm.

Solution:

The given parameters are

l=4cm and n=4

The formula for finding the area

=>

=>A = (4)² × 4/4tan(180/4)

=>A = 64/4 × 1

=>A = 16cm²