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


Formula length of a side :
Area of Polygon = {[ side ]² . N] / [4.Tan(π / N]}
Perimeter of Polygon = N . [side]

Formula radius (circumradius) :
Area of Polygon = ½ . R² . Sin[2.π / N]

Formula apothem [inradius] :
Area of Polygon = A² . N . Tan(π / N)
where A = R . Cos(π / N)

Formula apothem and length of a side :
Area of Polygon = [A x P] / 2
where A = side / [2 x Tan(π / N]

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


Example 

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

Smart Math Formula : Find the area.
Area = {[side]² . N] / [4.Tan[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.

Smart Math Formula : Find the perimeter.
Perimeter = [N .[side] = 4 . 2 = 8

Question : Find the area of a polygon with the given radius 2 and the number of sides is 5.
Smart Math Formula : Find the area.
Area = ½ . R² . Sin[2.π / N]
= [0.5] . 2² . Sin(2 . 3.14 / 5]
= 0.5 x 4 x Sin[6.28 / 5]
= 2 x Sin[1.26]
= 2 x 0.95
Area = 1.9.

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

Smart Math Formula : Find the apothem.
Apothem = R . Cos[π / N]
= 2 . Cos[3.14 / 5]
= 2 . Cos[0.63]
= 2 . 0.81
Apothem = 1.62.

Smart Math Formula : 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.

Smart Math Formula : 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.

Smart Math Formula : Find the perimeter.
Perimeter = [N x [side] = 4 x 2 = 8

Smart Math Formula : Find the area.
Area = [A x P] / 2
= [1 x 8] / 2
= 8 / 2
Area = 4.

That example be smart illustrates how to calculate the Area, Perimeter of a Polygon Smart Math Formula.