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The perimeter of any 2D figure is defined as the distance around the figure. You can find the perimeter of any plane figure (2D shape) by adding the length of each side of the figure. Since the perimeter of any figure is a linear measurement, it is expressed in terms of unit, such as mm, cm, m, ft, yd, in, etc.

Let’s learn what is the perimeter of a rectangle and how it is calculated.

## Triangle – A 2D Plane Figure

A triangle is a three-sided polygon formed by joining three non-collinear points known as the vertices (singular vertex). The line segments joining the three points (vertices) are called the sides (edges) of a triangle. The region formed between any of the two sides/edges of a triangle is called an angle of a triangle. There are three angles in a triangle whose sum is always $180^{\circ}$.

Triangles are broadly classified under two classifications

- Classification based on sides
**Scalene Triangle:**A triangle with no side equal**Isosceles Triangle:**A triangle with any two sides is equal**Equilateral Triangle:**A triangle with all three sides are equal

- Classification based on angles
**Acute Triangle:**A triangle where all three angles acute angles $\left(\lt 90^{\circ} \right)$**Obtuse Triangle:**A triangle where one of the angles is an obtuse angle $\left(\gt 90^{\circ} \right)$**Right Triangle:**A triangle where one of the angles is a right angle $\left(= 90^{\circ} \right)$

## What is the Perimeter of Triangle?

The perimeter of a triangle is the total length of all three sides of a triangle. Thus, the perimeter of a triangle is obtained by adding the length of the three sides of a triangle. Since the perimeter is a linear measure, therefore, the unit of the perimeter of a rectangle will be in metre, centimetre, inch, feet, etc.

$\text {Perimeter of a triangle = $ a + b + c$, where $a$, $b$, and $c$ are the sides of a triangle.

### Perimeter of Scalene Triangle

As the three sides in a scalene triangle are of different lengths, the perimeter of a scalene triangle is calculated by finding the sum of these three sides.

The perimeter of a scalene triangle is given by the formula Perimeter = $a + b + c$, where $a$, $b$, and $c$ are the length of the three different sides of a triangle.

### Examples

**Ex 1:** Find the perimeter of a triangle whose sides are $4 cm$, $8 cm$, and $6 cm$.

The three sides of a triangle are $a = 4 cm$, $b = 8 cm$ and $c = 6 cm$

$ a + b + c = 4 + 8 + 6 = 18$.

Perimeter of a triangle with sides $4 cm$, $8 cm$ and $6 cm$ is $18 cm$.

**Ex 2:** Find the length of the third side of a triangle whose perimeter is $36 cm$ and the measure of the two sides are $13 cm$ and $14 cm$.

In the question, $P = 36 cm$, and $a = 13 cm$, $b = 14 cm$

Perimeter $P = a + b + c => 36 = 13 + 14 + c => 36 = 27 + c => c = 36 – 27 = 9$

The measure of the third side of a triangle of perimeter $36 cm$ and lengths of the two sides $13 cm$ and $14 cm$ is $9 cm$.

### Perimeter of Isosceles Triangle

In an isosceles triangle, the length of any of the two sides is equal and the third side is of a different length. If the length of two equal sides is $a$ and that of the third unequal side is $b$, then the perimeter of an isosceles triangle is obtained by taking the sum of these three sides.

Perimeter = $a + a + b = 2a + b$.

Therefore, the perimeter of an isoscles triangle = $\text{Twice the equal sides} + \text{Unequal side}$

### Examples

**Ex 1:** Find the perimeter of an isosceles triangle whose two equal sides are of length $13 cm$ and that of the third side is $20$.

Length of two equal sides $a = 13 cm$

Length of third unequal side $b = 20 cm$

Perimeter of triangle = $2a + b = 2 \times 13 + 20 = 26 + 20 = 46 cm$

**Ex 2:** Find the length of the third side of an isosceles whose perimeter is $64 cm$ length of two equal sides is $24 cm$

Perimeter P = $64 cm$

Length of two equal sides $a = 24 cm$

$P = 2a + b => 64 = 2\times 24 + b => 64 = 48 + b => b = 64 – 48 => b = 16$

The measure of the third unequal side is $16 cm$.

### Perimeter of Equilateral Triangle

In the case of an equilateral triangle, the length of each side is equal. If the length of each side of an equilateral triangle is $a$, then the perimeter of an equilateral triangle is given by

Perimeter = $a + a + a = 3\times a$.

### Examples

**Ex 1:** Find the perimeter of an equilateral triangle whose measure of sides is $7 cm$.

Length of a side of an equilateral triangle $a = 7 cm$

Perimeter of an equilateral triangle $P = 3a = 3 \times 7 = 21 cm$.

**Ex 2:** Find the length of sides of an equilateral triangle whose perimeter is $24 cm$

Let the length of three equal sides of an equilateral triangle = $a$

Therefore, $3a = 24 => a = \frac {24}{3} => a = 8 cm$.

The length of each side of an equilateral triangle whose perimeter is $24 cm$ is $8 cm$.

### Perimeter of Right Triangle

A triangle that has one of the angles a right angle $\left( 90^{\circ}\right)$ is called a right triangle. If the length of the three sides of a right triangle is $a$, $b$, and $c$ (hypotenuse), then the perimeter of a triangle is calculated by finding the sum of these three sides. Therefore,

Perimeter = $a + b + c$.

Also, the sides of a right triangle are related to each other by the Pythagoras theorem.

$c^{2} = a^{2} + b^{2}$, which can also be written as $a^{2} = c^{2} – b^{2}$ or $b^{2} = c^{2} – a^{2}$. These relations on simplification give

$c = \sqrt{a^{2} + b^{2}}$, $a = \sqrt{c^{2} – b^{2}}$, and $b = \sqrt{c^{2} – a^{2}}$

If any one side of this triangle is not known, then we can use one of the above relations in place of that side and can find the perimeter of a triangle.

- When only sides $a$ and $b$ (Legs of right triangle or base and height of a right triangle) are known, then $P = a + b + \sqrt{a^{2} + b^{2}}$.
- When one side $a$ and hypotenuse $c$ are known, then $P = a + \sqrt{c^{2} – a^{2}} + c$.
- When one side $b$ and hypotenuse $c$ are known, then $P = \sqrt{c^{2} – b^{2}} + b + c$.

### Examples

**Ex 1:** Find the perimeter of a right triangle whose two perpendicular sides are of length $3 cm$ and $4 cm$ respectively.

Length of two perpendicular sides are $a = 3 cm$ and $b = 4 cm$.

$P = a + b + \sqrt{a^{2} + b^{2}} = 3 + 4 + \sqrt{3^{2} + 4^{2}} = 7 + \sqrt{9 + 16} = 7 + \sqrt{25} = 7 + 5 = 12 cm$

**Ex 2:** Find the perimeter of a right triangle whose hypotenuse is $13 cm$ and the length of one of the perpendicular sides is $12 cm$.

Length of hypotenuse of a right triangle $c = 13 cm$

Length of one of the perpendicular sides $a = 12 cm$

Perimeter $= a + \sqrt{c^{2} – a^{2}} + c = 12 + \sqrt{13^{2} – 12^{2}} + 13 = 12 + \sqrt{169 – 144} + 13 = 12 + \sqrt{25} + 13 = 12 + 5 + 13 = 30 cm$.

The perimeter of a right triangle whose hypotenuse is $13 cm$ and one of the perpendicular sides is $12 cm$ is $30 cm$.

### Perimeter of Isosceles Right Triangle

A triangle is called an isosceles right triangle when it is an isosceles triangle as well as a right triangle. If $a$ is the length of two equal sides and $b$ is the length of the hypotenuse, then the perimeter of a triangle is given by

$P = a + a + b = 2a + b$

Also, by Pythagoras Theorem, $b^{2} = a^{2} + a^{2} => b^{2} = 2a^{2}$ or $a^{2} = \frac {b^{2}}{2}$, which gives

$a = \frac {b}{\sqrt{2}}$ and $b = a\sqrt{2}$.

Therefore, the perimeter of an isosceles right triangle

- When only the length of equal sides is known is $P = 2a + a\sqrt{2} = a\left( 2 + \sqrt{2}\right )$
- When only the length of hypotenuse is known is $P = 2a + b = \frac {b}{\sqrt{2}} + b = b\left( \frac {1}{\sqrt{2}} + 1\right) = b\left( \frac {1 + \sqrt{2}}{\sqrt{2}}\right)$

### Examples

**Ex 1:** Find the perimeter of an isosceles right triangle whose length of equal sides is $4 cm$.

Length of equal sides $a = 4 cm$.

Perimeter of isosceles right triangle = $a\left( 2 + \sqrt{2}\right ) = 4\left( 2 + \sqrt{2}\right ) = 4\left( 2 + 1.4142\right ) = 4\times 3.4142 = 13.6568 = 13.66 cm$. (Rounded off to two decimal places).

**Ex 2:** Find the perimeter of an isosceles right triangle whose hypotenuse is of length $10 cm$.

Length of the hypotenuse of an isosceles right triangle $b = 10 cm$

Perimeter of isosceles right triangle = $b\left( \frac {1 + \sqrt{2}}{\sqrt{2}}\right) = 10\left( \frac {1 + \sqrt{2}}{\sqrt{2}}\right) = 10\left( \frac {1 + 1.4142}{1.4142}\right) = 10 \times \frac {2.4142}{1.4142} = 10 \times 1.7071 = 17.07 cm$

## Conclusion

The perimeter of a triangle is the length of the boundary formed by the three sides of a triangle. The perimeter of a triangle is calculated by finding the sum of the three sides of a triangle.

## Practice Problems

- Find the perimeter of a triangle whose sides are
- $4 cm$, $7 cm$ and $8 cm$
- $2 cm$, $6 cm$ and $5 cm$

- Find the perimeter of an isosceles triangle whose
- Equal sides are $6 cm$ and the third side is $8 cm$
- Equal sides are $7 cm$ and the third side is $12 cm$

- Find the perimeter of a right triangle whose two sides are
- $6 cm$ and $8 cm$
- $10 cm$ and $26 cm$

## Recommended Reading

- What is Length? (With Definition, Unit & Conversion)
- Weight – Definition, Unit & Conversion
- What is Capacity (Definition, Units & Examples)
- What is Time? (With Definition, Facts & Examples)
- What is Temperature? (With Definition & Units)
- Reading A Calendar
- What Are 2D Shapes – Names, Definitions & Properties
- Perimeter of Rectangle – Definition, Formula & Examples
- Perimeter of Square – Definition, Formula & Examples
- Area of Rectangle – Definition, Formula & Examples
- Area of Square – Definition, Formula & Examples

## FAQs

### What is the perimeter of a triangle in math?

The perimeter of a triangle is defined as the total length of its boundary. It is the sum of all three sides of the triangle.

### What is the formula for the perimeter of a triangle?

The perimeter of a triangle with measure of three sides as $a$, $b$ and $c$ is given by $P = a + b + c$.

### How do you find the perimeter of a triangle with three equal sides?

If three equal sides of a triangle is $a$, then the perimeter is given by $P = 3a$, since $a + a + a = 3a$.

### How do you find the perimeter of a triangle with two equal sides?

For an isosceles triangle with the measure of equal sides as $a$ and that of an equal side as $b$, the perimeter is given by $P = 2a + b$.