Polygon Tutorial

Polygon Definition:

A polygon is a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments.

Polygon Formula :

Using length of a side :

Area of Polygon = ((side)² * N) / (4Tan(π / N)) Perimeter of Polygon = N * (side)

Using radius (circumradius) :

Area of Polygon = ½ * R² * Sin(2π / N)

Using apothem (inradius) :

Area of Polygon = A² * N * Tan(π / N)

where

A = R * Cos(π / N)

Using apothem and length of a side :

Area of Polygon = (A * P) / 2

where

A = side / (2 * Tan(π / N))

where, N = Number of sides, A = Apothem, R = Radius, P = Perimeter

Polygon Image/Diagram

Polygon Example :

Case 1:

Find the area and perimeter of a polygon with the length 2 and the number of sides is 4.

Step 1:

Find the area.

Area	= ((side)² * N) / (4Tan(π / N))
	= ((2)² * 4) / (4 * Tan(3.14 / 4))
	= (4 * 4) / 4 * Tan(0.785)
	= 16 / 4 * 0.999
	= 16 / 3.996
Area	= 4.

Step 2:

Find the perimeter. Perimeter = (N * (side) = 4 * 2 = 8

Case 2:

Find the area of a polygon with the given radius 2 and the number of sides is 5.

Step 1:

Find the area.

Area	= ½ * R² * Sin(2π / N)
	= (0.5) * 2² * Sin(2 * 3.14 / 5)
	= 0.5 * 4 * Sin(6.28 / 5)
	= 2 * Sin(1.26)
	= 2 * 0.95
Area =	1.9.

Case 3:

Find the area of a polygon with the given radius 2 and the number of sides is 5 using Apothem.

Step 1:

Find the apothem.

Apothem	= R * Cos(π / N)
	= 2 * Cos(3.14 / 5)
	= 2 * Cos(0.63)
	= 2 * 0.81
Apothem =	1.62.

Step 2:

Find the area.

Area	= A² * N * Tan(π / N)
	= 1.62² * 5 * Tan(3.14 / 5)
	= 2.62 * 5 * Tan(0.63)
	= 13.1 * 0.73
Area	= 9.5.

Case 4:

Find the area of a polygon with the length 2 and the number of sides is 4 using Apothem.

Step 1:

Find the apothem.

Apothem	= side / (2 * Tan(π / N))
	= 2 / (2 * Tan(π / 4))
	= 2 / (2 * Tan(0.785))
	= 2 / (2 * 0.999)
	= 2 / 1.998
Apothem	= 1.

Step 2:

Find the perimeter. Perimeter = (N * (side) = 4 * 2 = 8

Step 3:

Find the area.


Area	= (A * P) / 2
	= (1 * 8) / 2
	= 8 / 2
Area	= 4.

The above example will clearly illustrates how to calculate the Area, Perimeter of a Polygon manually.

Polygon Tutorial