Polygons are two-dimensional figures which are made of straight line segments and each segment is connected with two other segments at each of its end points. The polygon could be regular or irregular. Polygons are called as regular when all the angles and sides are equal. If the angles and sides are not equal then such polygon's are known as irregular polygon.
 
The following figures show us some of the different shapes of polygon:

Polygon Picture

The total space within the boundary of the polygon is the area of polygon. The area of a polygon can be calculated using the length of the inradius, perimeter, number of sides and circumradius.

The area of a regular polygon is calculated in different circumstances. Here are few formulas to calculate area under those different circumstances.

1.When the length of a side is given:
Area of Polygon Length FormulaWhere,
S = Length of any side
N = Number of sides
tan = Tangent function in degrees


2. When the radius( distance from the center to the vertex) is given:
Area of Polygon Radius Formula
Where,
R = Radius
N = Number of sides
sin = sine function in degrees.


3. When the inradius/apothem(the perpendicular distance from center to side) is given:
Area of Polygon Inradius FormulaWhere,
A = Apothem or inradius
N = Number of sides
tan = Tangent function in degrees
The 4-sided polygon is called as quadrilateral.

Types of Quadrilateral:


1. Square: Square is a regular polygon and the best known quadrilateral. It consists equal sides and interior angles are all right angles.
The distance around the area is called as perimeter. Since all the four sides of a square is of same length, the perimeter of a square is given as, '4 side'. If s be the side of a square, then
Area of Square
Area of Square Formula
2. Rectangle: Rectangle is also the most commonly known quadrilateral like square. It consists of parallel and congruent opposite sides.

Rectangle Image

Area of Rectangle Formula

Where,
w = Width.
l = Length.

3. Parallelogram: A quadrilateral with parallel opposite sides is known as Parallelogram. It is the parent quadrilateral for rectangle, rhombus and square.
Parallelogram Image
Area of Parallelogram
Where,
b = Base
a = Height

4.Trapezoid:
The trapezoid is a quadrilateral which has 4 sides and with only one pair of parallel sides.

Area of Trapezoid
Area of Trapezoid
where,
b1= Length of 1st base
b2= Length of 2nd base
h = Height of trapezoid

5. Kite: Kite is a quadrilateral which has two pairs of equal sides. The interior angles of the opposite vertices of the kite are equal. The diagonals of kite intersect at the right angles.

Area of Kite
Area of Kite Formula
Where,
d1= Length of larger diagonal.
d2= Length of smaller diagonal.
Pentagon is a polygon with 5 sides and the sum of the interior angles is 540°. When all the sides and angles are equal, then the pentagon is called as regular, or else they are called as irregular. A regular pentagon has each interior angle of 108 degrees and exterior angle is of 72 degrees.
Area of Regular Pentagon
Area of Regular Pentagon = $a^2$ $\frac{n}{4}$ tan($\frac{\pi}{n}$
Where,
n = Number of sides
a = Length of any sides
tan = Tangent function in degrees.
If the all the sides and angles of a polygon are not equal then such polygon's are called as irregular polygon. Since they do not have equal sides and angles, its difficult to find the area of polygon using one single formula. So, in such case the polygon should divided into other polygons and then calculate the area of corresponding polygons using formulas. Later, these area's should added to find that area of that irregular polygon.

Irregular Polygon

The above figure is an irregular polygon. And in this polygon, we find 1 triangle and 1 rectangle, here we should first calculate the area of the triangle and the rectangle and then add up the two area's to find the area of irregular polygon.
A polygon placed inside the circle is called as inscribed polygon. All the vertices of the polygon lie on the circumference of the circle. All the sides in an inscribed polygon are known as chords.

Area of Inscribed Polygon

Area of Inscribed Polygon Formula

Perimeter of Inscribed Polygon
where, r - Radius of circle
n - Number of sides of a polygon
Below are the few problems based on regular Polygon:

Solved Examples

Question 1: Calculate the area of the polygon with 5 sides and length of side is 4 cm.
Solution:
 
Given: N = 5 and S = 4 cm
 
Here number of sides and the length of each side are given, So we will use the formula

$Area$ = $\frac{s^{2}N}{4tan(\frac{180}{N})}$

By substituting the values, we get

$Area$ = $\frac{4^{2}5}{4tan(\frac{180}{5})}$

$\frac{(16)5}{4tan(\frac{180}{5})}$

=  $\frac{5(16)}{4(0.726)}$

= $\frac{80}{2.904}$

= 27.54

Thus, the area of the polygon is 27.54 cm2
 

Question 2: Calculate the area of the polygon with the radius of 10 cm and 6 sides.
Solution:
 
Given:

R = 10 cm and N = 6

Here sides and the radius of the polygon are given, So lets use the formula 

$Area$ = $\frac{R^{2}Nsin(\frac{360^o}{N})}{2}$

By substituting the values, we get

$Area$ = $\frac{10^{2}6sin(\frac{360^o}{6})}{2}$

= $\frac{(100)6sin(\frac{360^o}{6})}{2}$

= $\frac{(600)sin 60^o}{2}$

= $\frac{519.61}{2}$

= 259.8

Thus, the area of the polygon is 259.8 cm2
 

Question 3: Calculate the area of the polygon with the inradius of 7cm and 4 sides.
Solution:
 
Given:

A = 7cm and N = 4

Here sides and the inradius of the polygon are given, So lets use the formula 

$Area$ =$A^{2}Ntan($$\frac{180}{N}$$)$

By substituting the values, we get

$Area$ =$7^{2}4tan($$\frac{180}{4}$$)$

 =$(49)(4)tan($$\frac{180}{4}$$)$

= (196)(1.000)

= 196
Thus, the area of the polygon is 196 cm2