Formula, Area Of Regular Polygons Examples - Cuemath

Area of Polygons

The area of a polygon is defined as the area that is enclosed by the boundary of the polygon. In other words, we say that the region that is occupied by any polygon gives its area. In this lesson, we will learn to determine the area of polygons and find the difference between the perimeter and area of polygons in detail.

1.	What is the Area of a Polygon?
2.	Difference Between Perimeter and Area of Polygons
3.	Area of Polygon Formulas
4.	Area of Polygons with Coordinates
5.	FAQs on Area of Polygons

What is the Area of a Polygon?

The definition of the area of a polygon is the measure of the area that is enclosed by it. As polygons are closed plane shapes, thus, the area of a polygon is the space that is occupied by it in a two-dimensional plane. The unit of the area of any polygon is always expressed in square units. Observe the following figure which shows the area of a polygon on a two-dimensional plane.

Difference Between Perimeter and Area of Polygons

The perimeter and area of polygons are both measurable values that depend on the length of sides of the polygon. In order to differentiate between both of them, it is necessary to understand the basic difference between perimeter and area. Observe the table given below to understand this difference better.

Criteria of Difference	Perimeter of Polygon	Area of Polygon
Definition	It is defined as the total length of the boundary of the polygon which can be obtained by adding the length of all its sides.	It is defined as the region or space enclosed by any polygon.
Formula	The perimeter of Polygon = Length of Side 1 + Length of Side 2 + ...+ Length of side N (for an N sided polygon)	The area of polygons can be found by different formulas depending upon whether the polygon is a regular or an irregular polygon.
Unit	The unit of the perimeter of polygons is expressed in meters, centimeters, inches, feet, etc.	The unit of the area of polygons is expressed in (meters)2, (centimeters)2, (inches)2, (feet)2, etc.

The similarity between the calculation of perimeter and area of a polygon is that both depend on the length of the sides of the shape and not on the interior angles or the exterior angles of the polygon.

Area of Polygon Formulas

A polygon can be categorized as a regular or an irregular polygon based on the length of its sides. Thus, this differentiation also brings a difference in the calculation of the area of polygons. The area of some commonly known polygons is given as:

• Area of triangle = (1/2) × base × height We can also find the area of a triangle if the length of its sides is known by using Heron's formula which is, Area = \(\sqrt{s(s-a)(s-b)(s-c)}\), where s = Perimeter/2 = (a + b + c)/2, a, b, and c are the length of its sides.
• Area of rectangle = length × width
• Area of parallelogram = base × height
• Area of trapezium = (1/2) × (sum of lengths of its parallel sides or bases) × height
• Area of rhombus = (1/2) × (product of diagonals)

In order to calculate the area of a polygon, it must be first known whether the given polygon is a regular polygon or an irregular polygon.

Area of Regular Polygons

A regular polygon is a polygon that has equal sides and equal angles. Thus, the technique to calculate the value of the area of regular polygons is based on the formulas associated with each polygon. Let us have a look at the formulas of some commonly used regular polygons:

Names of Regular Polygon	Area of Regular Polygon
Equilateral Triangle	Area = (√3 ×(length of a side)2)/4
Square	Area = (length)2
Regular Pentagon	Area = 5/2 × side length × length of the apothem OR Area =\(\dfrac{1}{4} × \sqrt{5(5+2√5)} × (side)^2\)
Regular Hexagon	Area = [3√3 ×(length of a side)2]/2

In order to determine the area of a regular polygon, if the number of its sides are known, is given by:

• Area of regular polygon = (number of sides × length of one side × apothem)/2, where the length of apothem is given as the \(\dfrac{l}{2\tan(\dfrac{180}{n})}\), where l is the side length and n is the number of sides of the regular polygon.
• In terms of the perimeter of a regular polygon, the area of a regular polygon is given as, Area = (Perimeter × apothem)/2, in which perimeter = number of sides × length of one side

Example: Find the area of a regular pentagon whose side is 7 inches long. Solution: Given the length of one side = 7 inches. Hence, the area of the regular pentagon is given as A = \(\dfrac{1}{4} × \sqrt{5(5+2√5)} × (side)^2\) ⇒ A = \(\dfrac{1}{4} × \sqrt{5(5+2√5)} × (7)^2\) ⇒ A = 84.3 square inches

Thus, the area of the regular pentagon is 84.3 square inches.

Area of Irregular Polygons

An irregular polygon is a plane closed shape that does not have equal sides and equal angles. Thus, in order to calculate the area of irregular polygons, we split the irregular polygon into a set of regular polygons such that the formulas for their areas are known. Consider the example given below.

The polygon ABCD is an irregular polygon. Thus, we can divide the polygon ABCD into two triangles ABC and ADC. The area of the triangle can be obtained by: Area of polygon ABCD = Area of triangle ABC + Area of triangle ADC

Area of Polygons with Coordinates

The area of polygons with coordinates can be found using the following steps:

• Step 1: First we find the distance between all the points using the distance formula, D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\)
• Step 2: Once, the dimensions of the polygons are known find whether the given polygon is a regular polygon or not.
• Step 3: If the polygon is a regular polygon we use the formula, area of regular polygon = (number of sides × length of one side × apothem)/2, where the length of apothem is given as the (length of one side)/(2 ×(tan(180°/number of sides))). If the polygon is an irregular polygon, it is to be divided into several regular polygons to find the area.

Example: What is the area of the polygon formed by the coordinates A(0,0), B(0, 2), C(2, 2), and D(2, 0)? Solution: On plotting the coordinates A(0,0), B(0, 2), C(2, 2), and D(2, 0) on an XY plane and joining the dots we get,

It can be seen, the obtained figure shows a four-sided polygon. In order to understand whether it is a regular polygon or not, we find the distance between all the points.

Length of AB = \(\sqrt{({0 - 0})^2 + ({2 - 0})^2}\) = 2 units Length of BC = \(\sqrt{({2 - 0})^2 + ({2 - 2})^2}\) = 2 units Length of CD = \(\sqrt{({2 - 2})^2 + ({0 - 2})^2}\) = 2 units Length of DA = \(\sqrt{({0 - 2})^2 + ({0 - 0})^2}\) = 2 units

Now that we know the length of all sides of the given polygon is the same, it shows, it is a square. Thus, the area of the polygon ABCD is given as A = (length)2 = (2)2 = 4 square units.

Hence, the area of the polygon with coordinates (0,0), (0, 2), (2, 2), and (2, 0) is 4 square units.

Important Notes

• If the length of one side is given, it possible to find the area of the regular polygon by finding apothem.
• Apothem falls on the midpoint of a side at the right angle dividing it into two equal parts.
• An Equilateral triangle is a regular polygon with 3 sides, while a square is a regular polygon with 4 sides. Hence, they are not prefixed as regular ahead of the shape name.