Perimeter Of Polygon - Steps, Formula, Solved Examples - Cuemath

Perimeter of Polygon

The perimeter of a polygon is defined as the sum of the length of the boundary of the polygon. In other words, we say that the total distance covered by the sides of any polygon gives its perimeter. In this lesson, we will learn to find the perimeter of polygons, and find the difference between the area and perimeter of the polygons in detail.

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

What is the Perimeter of Polygon?

The perimeter of a polygon is the measure of the total length of the boundary of the polygon. As polygons are closed plane shapes, thus, the perimeter of the polygons also lies in a two-dimensional plane. The perimeter of a polygon is always expressed in linear units like meters, centimeters, inches, feet, etc. For example, if the sides of a triangle are given as 4 cm, 6 cm, and 7 cm, then its perimeter will be, 4 + 6 + 7 = 17 cm. This basic formula applies to all polygons.

Difference Between Area and Perimeter of Polygon

The area and perimeter of polygons can be calculated if the lengths of the sides of the polygon are known. The following table shows the difference between the area and perimeter of polygons.

Criteria of Difference	Area of Polygon	Perimeter of Polygon
Definition	The space enclosed by any polygon is known as its area.	The perimeter of a polygon is defined as the total length of its boundary.
Formula	The area of polygons is calculated using different formulas depending on the type of polygon. For example, the area of a square = a2, where 'a' is its side length; the area of a rectangle = length × width,	The basic formula used to find the perimeter of a polygon is, Perimeter = sum of all sides.
Unit	The area of polygons is expressed in square units like meters2, centimeters2, inches2, feet2, etc.	The perimeter is expressed in linear units like meters, centimeters, inches, feet, etc.

There is one similarity between the area and perimeter of a polygon. Both depend directly on the length of the sides of the shape and not directly on the interior angles or the exterior angles of the polygon.

Formula for Perimeter of Polygon

We can categorize a polygon as a regular or irregular polygon based on the length of its sides. The perimeter formula of some known polygons is given as follows:

• Perimeter of a triangle = a + b + c, where, a, b, and c are the length of its sides.
• Perimeter of a rectangle = 2 × (length + width)

Before calculating the perimeter of the polygon, we first find out whether the given polygon is a regular polygon or an irregular polygon. After that, the appropriate formula is used to find the perimeter of the polygon.

Perimeter of Regular Polygons

A polygon that is equilateral and equiangular is known as a regular polygon. Thus, we calculate the perimeter of regular polygons using the formulas associated with each polygon. The formulas of some commonly used regular polygons are:

Names of Regular Polygon	Perimeter of Regular Polygon
Equilateral Triangle	3 × (length of one side)
Square	4 × (length of one side)
Regular Pentagon	5 × (length of one side)
Regular Hexagon	6 × (length of one side)

Therefore, the formula to find the perimeter of a regular polygon is: Perimeter of regular polygon = (number of sides) × (length of one side)

Example: Find the perimeter of a regular hexagon whose each side is 6 inches long.

Solution: Given, the length of one side = 5 inches and the number of sides = 6 (as it is a hexagon).

Thus, the perimeter of the regular hexagon = (number of sides) × (length of one side) = (6 × 5) = 30 inches. Therefore, the perimeter of the regular hexagon is 30 inches.

Perimeter of Irregular Polygons

Polygons that do not have equal sides and equal angles are referred to as irregular polygons. Thus, in order to calculate the perimeter of irregular polygons, we add the lengths of all sides of the polygon.

Example: Find the perimeter of the given polygon.

Solution: As we can see, the given polygon is an irregular polygon since the length of each side is different (AB = 7 units, BC = 8 units, CD = 3 units, and AD = 5 units)

Thus, the perimeter of the irregular polygon will be the sum of the lengths of all its sides. The perimeter of ABCD = AB + BC + CD + AD ⇒ Perimeter of ABCD = (7 + 8 + 3 + 5) = 23 units

Therefore, the perimeter of ABCD is 23 units.

Perimeter of Polygon with Coordinates

The perimeter of a polygon with coordinates can be found using the following steps:

• Step 1: Find the distance between all the points using the distance formula, D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\), where, \((x_1, y_1)\) and \((x_2, y_2) \) are the coordinates.
• Step 2: Once the dimensions of the polygon are known, we need to find whether the given polygon is a regular polygon or not.
• Step 3: If the polygon is a regular polygon we use the formula, perimeter of regular polygon = (number of sides) × (length of one side); while if the polygon is an irregular polygon we just add the lengths of all sides of the polygon.

Example: What is the perimeter of the polygon formed by the coordinates A(0,0), B(0, 3), C(3, 3), and D(3, 0)?

Solution: On plotting the coordinates A(0,0), B(0, 3), C(3, 3), and D(3, 0) on an XY plane and joining the dots we get a four-sided polygon as shown below.

In order to understand whether it is a regular polygon or not, we need to find the distance between all the points using the distance formula, D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\), where, \((x_1, y_1)\) and \((x_2, y_2) \) are the coordinates. After substituting the values in the formula, the length of sides AB, BC, CD and DA can be calculated as shown below.

• Length of AB = \(\sqrt{({0 - 0})^2 + ({3 - 0})^2}\) = 3 units. This was calculated with \(x_1\) = 0, \(x_2\) = 0, \( y_1\) = 0, \( y_2\) = 3
• Length of BC = \(\sqrt{({3 - 0})^2 + ({3 - 3})^2}\) = 3 units. This was calculated with \(x_1\) = 0, \(x_2\) = 3, \( y_1\) = 3, \( y_2\) = 3
• Length of CD = \(\sqrt{({3 - 3})^2 + ({0 - 3})^2}\) = 3 units. This was calculated with \(x_1\) = 3, \(x_2\) = 3, \( y_1\) = 3, \( y_2\) = 0
• Length of DA = \(\sqrt{({0 - 3})^2 + ({0 - 0})^2}\) = 3 units. This was calculated with \(x_1\) = 3, \(x_2\) = 0, \( y_1\) = 0, \( y_2\) = 0

Now, we know that the length of all sides of the given four-sided polygon is the same. This shows that it is a square. Thus, the perimeter of the polygon ABCD (square) can be calculated with the formula, Perimeter = number of sides) × (length of one side). After substituting the values in the formula, we get, perimeter = 4 × 3 = 12 units. Hence, the perimeter of the polygon with coordinates (0,0), (0, 3), (3, 3), and (3, 0) is 12 units.