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# Perimeter of a Polygon(With Formula & Examples)

September 3, 2022

The total distance/length of the sides of a closed 2D figure is known as the perimeter. It is calculated as Perimeter = Sum of all sides”. The unit of the perimeter of any polygon will remain the same as the unit of its respective sides. If the sides are given in different units, then the first step is to convert them to the same unit and then find the perimeter.

Let’s learn how to find the perimeter of polygons.

## Polygon – A 2D Plane Figure

A polygon is a  two-dimensional closed shape bounded with straight sides known as its sides or edges. It does not have curved sides. The points where two sides meet are the vertices (singular vertex) of a polygon. Each vertex contains two sides forming an angle.

The polygons are named based on the number of sides present in them. Some of the common polygons are

Note:

• The suffix ‘gon’ in a polygon is a Greek word meaning side
• The ‘poly’ is also a Greek word meaning many

## Regular and Irregular Polygons

A polygon is broadly categorized as a regular and irregular polygon based on the length of its sides. As the name suggests the length of all sides is equal and also the measure of each angle is equal. Therefore, one can use a fixed or a general formula to find the perimeter of a polygon by using either of the following

• number of sides
• measure of an interior angle
• measure of an exterior angle

In the case of an irregular polygon, the length of the sides is different, and also the measure of angles is different. Therefore, to find the perimeter of an irregular polygon length of all the sides should be known.

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## Perimeter of Irregular Polygon

As discussed above 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.

### Examples

Ex 1: Find the perimeter of a polygon with sides $4 cm$, $8 cm$, $10 cm$, and $9 cm$.

Let the four sides of a polygon be $a = 4 cm$, $b = 8 cm$, $c = 10 cm$ and $d = 9 cm$

Perimeter = $a + b + c + d = 4 + 8 + 10 + 9 = 31 cm$

Note: The polygon is an irregular quadrilateral.

Ex 2: Find the perimeter of the given hexagon

The length of the six sides of the hexagon are $a = 4 cm$, $b = 18 cm$, $c = 7 cm$, $d = 9 cm$, $e = 10 cm$ and $f = 15 cm$.

The perimeter of the hexagon = $a + b + c + d + e + f = 4 + 18 + 7 + 9 + 10 + 15 = 63 cm$.

## Perimeter of Regular Polygon

As discussed above a regular polygon has sides of the same length and angles of the same measure, so it becomes easy to formulate a fixed formula for finding the perimeter of a polygon.

You can find the perimeter of a regular polygon if you know either of the following along with the length of a side of a polygon.

• number of sides
• measure of an interior angle
• measure of an exterior angle

### Perimeter of Regular Polygon With Known Number of Sides

For a polygon with $n$ sides, and the length of each side $l$, the formula for perimeter is given by Perimeter = $n \times l$. Let’s consider some examples to understand it.

### Examples

Ex 1: Find the perimeter of an equilateral triangle of side $9 in$.

Length of each side $l = 9 in$

Number of sides in an equilateral triangle $n = 3$

Therefore, perimeter = $n \times l = 3 \times 9 = 27 in$.

Ex 2: Find the perimeter of a regular heptagon of side $11 cm$.

Length of each side $l = 11 cm$.

Number of sides in a regular heptagon $n = 7$.

Therefore, perimeter = $n \times l = 7 \times 11 = 77 cm$.

Ex 3: What is the name of a regular polygon with the length of each side $15 in$ and a perimeter of $120 in$?

Let the number of sides = $n$

Length of each side $l = 15 in$

Perimeter $P = 120 in$

$P = n \times l => 120 = n \times 15 => n = \frac {120}{15} = 6$

Therefore, the number of sides in a polygon is $6$, and hence, it is a regular hexagon.

### Perimeter of Regular Polygon With Known Measure of Interior Angle

If the measure of each of the interior angles of a polygon is $i$ and the number of sides in a polygon is $n$, then

$i = \frac {\left(n – 2 \right) \times 180}{n} => \frac {ni}{180} = n – 2$

$=>n – \frac {ni}{180} = 2 => n(1 – \frac {i}{180}) = 2$

$=>n\frac {180 – i}{180} = 2 => n = \frac {360}{180 – i}$

And if the length of each side is $l$, then the perimeter of a polygon is given by $P = \frac {360l}{180 – i}$

### Examples

Ex 1: Find the perimeter of a regular polygon of interior angle $108^{\circ}$ and the length of each side $7 in$.

The perimeter is given by $P = \frac {360l}{180 – i}$

Here, $i = 108^{\circ}$ and $l = 7$

Substituting the respective values, in $P = \frac {360l}{180 – i}$ we get $P = \frac {360 \times 7}{180 – 108}$

$= \frac {2520}{72} = 35 in$

Note: The polygon having the measure of each interior angle $108^{\circ}$ is a pentagon (number of sides = $5$).

Ex 2: What is the measure of each side of a regular polygon, if its perimeter is $88 cm$ and the interior angle is $135^{\circ}$.

In this problem, perimeter $P = 88 cm$ and measure of each interior angle $i = 135^{\circ}$.

Substituting the respective values, in $P = \frac {360l}{180 – i}$ we get Substituting the respective values, in $88 = \frac {360l}{180 – 135} => 88 = \frac {360l}{45} => 360l = 88 \times 45$

$=> 360l = 3960 => l = \frac {3960}{360} => l = 11$.

Therefore, the number of sides in a polygon is $11$.

### Perimeter of Regular Polygon With Known Measure of Exterior Angle

If the measure of each exterior angle of a regular polygon is $e$, then the number of sides of a polygon is $\frac {360}{e}$ and the perimeter of such a polygon can be calculated by using the formula $P = \frac {360l}{e}$, where the measure of each side is $l$.

### Examples

Ex 1: Find the perimeter of a polygon of each side $12 cm$ and exterior angle $90^{\circ}$.

Length of each side $l = 12 cm$

And the measure of each exterior angle $e = 90^{\circ}$

The perimeter of a polygon is given by $P = \frac {360l}{e}$.

Substituting the respective values, we get $P = \frac {360 \times 12}{90} = \frac {4320}{90} = 48 cm$.

Note: A regular polygon whose exterior angle is $90^{\circ}$ is a square.

Ex 2: Find the measure of each side of a regular polygon whose exterior angle is $45^{\circ}$ and perimeter is $72 m$.

Measure of each exterior angle $e = 45^{\circ}$ and perimeter $P = 72 m$.

Substituting the values in $P = \frac {360l}{e}$, we get $72 = \frac {360l}{45} => l = \frac {72 \times 45}{360} => l = 9$.

The measure of each side is $9 m$.

Note: A regular polygon with the measure of each exterior angle $45^{\circ}$ is an octagon.

## Perimeter of Polygon with Coordinates

To find the perimeter of a polygon whose coordinates are known, you use the distance formula to find the length of each side between the vertices of a polygon.

The distance formula is $D = \sqrt{\left(x_{2} – x_{1} \right)^{2} + \left(y_{2} – y_{1} \right)^{2}}$.

After finding the distances (lengths) between each pair of vertices, add the lengths to get the perimeter of a polygon.

$P = l_{1} + l_{2} + l_{3} + … + l_{n}$, where $l_{1}$, $l_{2}$, $l_{3}$, …, $l_{n}$ are the lengths of sides of a polygon.

Note: Using this formula you can find the perimeter of regular as well as irregular polygons.

### Examples

Ex 1: Find the perimeter of a triangle whose coordinates of the vertices are $\left(3, 9 \right)$, $\left(2, 2 \right)$, and $\left(7,5 \right)$.

The three vertices of a triangle are

$A = \left(3, 9 \right)$, $B = \left(2, 2 \right)$, and $C = \left(7,5 \right)$.

Length of side $AB$ is $\sqrt{\left(2 – 3 \right)^{2} + \left(2 – 9 \right)^{2}} = \sqrt{\left(-1 \right)^{2} + \left(-7 \right)^{2}} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2}$.

Length of side $BC$ is $\sqrt{\left(7 – 2 \right)^{2} + \left(5 – 2 \right)^{2}} = \sqrt{5^{2} + 3^{2}} = \sqrt{25 + 9} = \sqrt{34}$.

Length of side $CA$ is $\sqrt{\left(3 – 7 \right)^{2} + \left(9 – 5 \right)^{2}} = \sqrt{\left(-4 \right)^{2} + 4^{2}} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$.

Therefore, perimeter = $5\sqrt{2} + \sqrt{34} + 4\sqrt{2} = \left (9\sqrt{2} + \sqrt{34} \right) unit$

## Conclusion

A polygon is a  two-dimensional closed shape bounded with straight sides known as its sides or edges. You can find the perimeter of a regular polygon if you know the length of the side and either the interior angle or the exterior angle of a polygon.

## Practice Problems

1. Find the perimeter of a polygon whose length of each side and interior angles are
1. Length of each side = $3 cm$, Interior angle = $90^{\circ}$
2. Length of each side = $7 cm$, Interior angle = $120^{\circ}$
3. Length of each side = $3 cm$, Interior angle = $108^{\circ}$
2. Find the perimeter of a polygon whose length of each side and exterior angles are
1. Length of each side = $2 cm$, Interior angle = $60^{\circ}$
2. Length of each side = $11 cm$, Interior angle = $30^{\circ}$
3. Length of each side = $8 cm$, Interior angle = $36^{\circ}$

## FAQs

### Does a polygon have a perimeter?

Polygons have at least three sides and three angles. A polygon cannot have any curved sides. Polygons can be categorized as regular or irregular. Regular polygons have equal side lengths and identical interior angle measures.

### How do you calculate the sides of a polygon if its exterior angle is known?

To calculate the number of sides of the polygon, divide $360$ by the measure of the exterior angle. For example, if the exterior angle is $60^{\circ}$, then dividing $360$ by $60$ equals $6$, which is the number of sides the polygon has.

### How do you calculate the sides of a polygon if its interior angle is known?

To calculate the number of sides of the polygon, use the formula $n = \frac {360}{180 – i}$, where $i$ is the measure of each interior angle. For example, if the interior angle is $135^{\circ}$, then the number of sides will be \frac {360}{180 – 135} = 8\$.

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