How to calculate the Area, Perimeter of a Polygon manually

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 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.
Source : http://www.easycalculation.com/area/learn-polygon.php