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.

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.

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.

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.

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:

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.

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.

What is the Perimeter of Polygon?

The perimeter of a polygon is defined as the total length of the boundary of the polygon in a two-dimensional plane. The perimeter of a polygon is expressed in linear units like meters, centimeters, inches, feet, etc.

How to Find the Perimeter of a Polygon?

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

• Step 1: Find whether the given polygon is a regular polygon or not.
• Step 2: If it is a regular polygon, the perimeter can be calculated using the formula, Perimeter of regular polygon = (number of sides) × (length of one side). In case, if it is an irregular polygon, then its perimeter can be calculated by adding the lengths of all its sides.
• Step 3: Once the perimeter of the polygon is obtained, we need to mention the unit along with the value of the perimeter.

What is the Difference Between the Area and Perimeter of Polygons?

The perimeter of a polygon is the total length of its boundary, whereas, the area of a polygon is the space enclosed by the polygon. We can find the perimeter of a polygon by adding the length of all its sides. The area of a polygon is calculated by using the appropriate formulas or by reducing the polygon into smaller regular polygons. The area of a polygon is always expressed in square units, like meter², centimeter², while the perimeter of a polygon is always expressed in linear units like meters, inches, and so on.

How to Find the Perimeter of Polygons with Vertices?

We can find the perimeter of polygons with given vertices using the following steps:

• Step 1: First, we need to calculate 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 of the vertices.
• Step 2: Once the dimensions of the polygons are known, we need to check whether the given polygon is a regular polygon or an irregular polygon.
• Step 3: The perimeter of a regular polygon can be found by using the formula, perimeter of regular polygon = (number of sides) × (length of one side), whereas, if the polygon is an irregular one, we can simply add the lengths of all its sides.

How to Find the Perimeter of Regular Polygons?

The perimeter of regular polygons can be found using the following steps:

• Step 1: Count the number of sides of the polygon.
• Step 2: Note the length of one side.
• Step 3: Use the values obtained in Step 1 and Step 2 to find the value of perimeter using the formula, Perimeter of a regular polygon = (number of sides) × (length of one side).

How to Find the Perimeter of Irregular Polygons?

In order to calculate the perimeter of an irregular polygon we use the following steps:

• Step 1: Note the length of each side of the given polygon.
• Step 2: Once the length of all the sides is obtained, the perimeter is found by adding all the sides.

Is the Perimeter of a Regular Polygon Directly Proportional to the Length of Side?

Yes, the perimeter of a regular polygon is directly proportional to its side length. We know that the perimeter of a regular polygon is calculated by the formula, Perimeter = (number of sides) × (length of one side). Thus, if the length of the side is increased, the value of the perimeter also increases. For example, a square with a side length of 4 units will have a larger perimeter as compared to a square with a side length of 2 units.

How to Find the Missing Side Length When the Perimeter of Polygon is Given?

We can find the missing side length when the perimeter of the polygon is given in the following way:

• Step 1: Find whether the given polygon is a regular polygon or not.
• Step 2: If the given polygon is a regular polygon, then we use the formula, Perimeter of regular polygon = (number of sides) × (length of one side) to find the missing side length. In case, if the given polygon is an irregular polygon, then we add the lengths of all the given sides and subtract it from the perimeter to get the missing side.

What is the Formula of the Perimeter of Polygon?

The formula that is used to calculate the perimeter of a polygon is simple to understand because 'perimeter' means the sum of the length of all its sides and hence, the formula is expressed as, Perimeter = Sum of the sides. If it is a regular polygon, it means that all the sides are equal. In that case, to make things easier, the formula is expressed as, Perimeter = number of sides × length of one side.

How to Find the Perimeter of Polygons with Coordinates?

The perimeter of polygons with coordinates can be calculated by using the following steps.

• First, the length of the sides of the polygon can be calculated using the distance formula. The given coordinates, \((x_1, y_1)\) and \((x_2, y_2) \) are substituted in the distance formula, D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\).
• After the length is known, we should find out if the polygon is a regular polygon or an irregular one. Accordingly, the formula for the perimeter is used to calculate the perimeter.
• If it is a regular polygon, the formula that is used is, Perimeter = number of sides × length of one side. If it is an irregular polygon, then the sides can be added to find the perimeter using the formula, Perimeter = Sum of the sides.