In this article, we will see the definition of the perimeter of a triangle, its formula, and how to solve the perimeter of a triangle when points are given or an image is given, or when sides are missing.

## Definition of the Perimeter of a Triangle

The **Perimeter **of a triangle is the sum of all of its three sides.

The definition is the very same for the perimeter of every other polygon: the sum of all the sides. The perimeter is explicitly dependent on the sides of the triangle and independent of the sides. So if one is given the measure of all the sides of the triangle, the perimeter is simply the sum of all these sides. This can be further divided into three types: perimeter of a Scalene, Isosceles and Equilateral triangle.

## Formula for the Perimeter of a Scalene Triangle

Recall that a scalene triangle is a triangle whose all sides are different in length.

Let *a, b *and *c *be the length of all the sides of a scalene triangle, then according to the definition, the perimeter is given by:

$P=a+b+c$,

where *P *denotes the perimeter.

Find the perimeter of a triangle whose sides are of length a = 4, b = 5 and c = 7 units.

**Solution:**

The perimeter is given by the following:

$P=a+b+c\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=4+5+7\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=16units$

### Formula for the Perimeter of an Isosceles Triangle

An isosceles triangle is a triangle in which the length of two sides is the same and the third is different.

Let *a *be the length of each of the equal sides and the length of the third side be *b*, then the perimeter of the triangle will be:

$P=2a+b$.

It is just a special case of a scalene triangle for *a = c*.

Find the perimeter of an isosceles triangle whose two sides are of length 4 units each and the third side is 5 units.

**Solution:**

The perimeter of an isosceles triangle is given by:

$P=2a+b$.

where a = 4 and b = 5 are given:

$P=2\times 4+5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=8+5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=13$

Hence, the perimeter of this triangle is 13 units.

### Formula for the Perimeter of an Equilateral Triangle

An equilateral triangle is a triangle whose all sides are of the same length.

Let *a *be the length of each of the sides, then the perimeter is given by:

$P=3a$

Hence, the perimeter of an equilateral triangle is thrice the length of each of its sides. Again, an equilateral triangle is just a specific case of a scalene triangle.

Find the perimeter of an equilateral triangle whose length of each side is 3 units.

**Solution:**

We can use the formula to find the perimeter of an equilateral triangle:

$P=3a\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=3\times 3\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=9$

Hence the perimeter of the triangle is 9 units.

## Perimeter when points are given

Suppose that we are not directly given the lengths of the sides of a triangle, rather we are given the coordinates of its vertices. It is especially the case when a **triangle is inscribed within a cartesian plane**. The coordinates help to locate the triangle.

To find the perimeter when the coordinates of its vertices are given, we need to find the length of the individual sides somehow. To do so, we can calculate the distance between the vertices using the **distance formula**, and the line segments formed when we connect the vertices are the sides of the triangle itself.

Hence, the length of the line segments joining the vertices will be the same as the length of the sides of the triangle formed.

Let A, B and C be the vertices of the triangle and the coordinates be $\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right)\mathrm{and}\left({x}_{3},{y}_{3}\right)$ where all the points are distinct. The sides of the triangle ABC will be AB, BC and AC.

Using the distance formula:

$d=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$

So now that we have the length of the sides of the triangle in terms of the coordinates of the vertices, we can use the perimeter formula:

$P=\sqrt{{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}}+\sqrt{{\left({x}_{2}-{x}_{3}\right)}^{2}+{\left({y}_{2}-{y}_{3}\right)}^{2}}+\sqrt{{\left({x}_{3}-{x}_{1}\right)}^{2}+{\left({y}_{3}-{y}_{1}\right)}^{2}}$

Thus, we have found a formula for calculating the perimeter of the triangle whose vertices are given.

Find the perimeter of the triangle whose vertices are located at A(–3, 1), B(2, 1) and C(2, –1).

**Solution:**

In order to find the length of the perimeter, we need to find the length of the respective sides and we can do so by using the distance formula for all the three vertices.

For the first side AB:

$AB=\sqrt{{\left(-3-2\right)}^{2}+{\left(1-1\right)}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=5$

For the second side, BC:

$BC=\sqrt{{\left(2-2\right)}^{2}+{\left(1+1\right)}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=2$

And for the third side, AC:

$AC=\sqrt{{\left(-3-2\right)}^{2}+{\left(1+1\right)}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\sqrt{29}$

And now the perimeter can be calculated by adding all these sides:

$P=AB+BC+AC\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}P=5+2+\sqrt{29}=7+\sqrt{29}$

Hence the perimeter of the triangle whose vertices are A(–3, 1), B(2, 1) and C(2, –1) is $7+\sqrt{29}$ units.

## Perimeter of a Triangle with Missing sides

Sometimes, not all sides are given. Most of the time, the length of two of the sides and the angle between them is given. Using just this information, we have to find the perimeter of the triangle.

Again, using the given information, we need to find the length of the missing side. If one is given the angle between two of the known sides, cosine law is used to determine the length of the missing side.

Let A, B and C be the three angles of a triangle and the sides opposite to them have length respectively.

Suppose that we are given the length of two sides as *a, b, *and the angle between them is given as C.

Hence, we need to find the third side *c* using the given information, here we make use of the cosine law to determine the missing side. Recall that the **cosine law** is given as follows:

${c}^{2}={a}^{2}+{b}^{2}-2ab\mathrm{cos}C\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}c=\sqrt{{a}^{2}+{b}^{2}-2ab\mathrm{cos}C}$

where the only unknown is *c *and so we can calculate it quite easily. A very specific case would be a right-angled triangle where the cosC term would vanish since C = π/2 and the cosine law would simply yield the Pythagoras’ Theorem.

Now using the formula for the perimeter, of a triangle, we get:

$P=a+b+c$

Substituting for *c *we get:

$P=a+b+\sqrt{{a}^{2}+{b}^{2}-2ab\mathrm{cos}C}$

Find the perimeter of a triangle whose two sides are of length 4 and 5 units, and the angle between those sides is $\frac{\pi}{3}$ radian.

**Solution:**

Let us label the two sides as *a *and *b *respectively, so that *a = 4 *and *b = 5* , and the angle be *C*, we need to find the length of the remaining side in order to find the perimeter.

Using the cosine law here:

$c=\sqrt{{a}^{2}+{b}^{2}-2ab\mathrm{cos}C}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{c}^{2}=16+25-2.4.5\xb7\frac{1}{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}c=\sqrt{41-20}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}c=\sqrt{21}$

Therefore, the perimeter is given by:

$P=a+b+c\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}P=4+5+\sqrt{21}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}P=9+\sqrt{21}$

Thus the perimeter of the given triangle is $9+\sqrt{21}$ units.

## The Perimeter of a Triangle on a Graph

Suppose that we are not directly given the lengths of the sides of a triangle, rather we are given the coordinates of its vertices. It is especially the case when a **triangle is inscribed within a cartesian plane**. The coordinates help to locate the triangle.

To find the perimeter when the coordinates of its vertices are given, we need to find the length of the individual sides somehow. To do so, we can calculate the distance between the vertices using the **distance formula**, and the line segments formed when we connect the vertices are the sides of the triangle itself.

Hence, the length of the line segments joining the vertices will be the same as the length of the sides of the triangle formed.

Let A, B and C be the vertices of the triangle and the coordinates be $\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right)\mathrm{and}\left({x}_{3},{y}_{3}\right)$ where all the points are distinct. The sides of the triangle ABC will be AB, BC and AC.

Using the distance formula:

$d=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$

So now that we have the length of the sides of the triangle in terms of the coordinates of the vertices, we can use the perimeter formula:

$P=\sqrt{{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}}+\sqrt{{\left({x}_{2}-{x}_{3}\right)}^{2}+{\left({y}_{2}-{y}_{3}\right)}^{2}}+\sqrt{{\left({x}_{3}-{x}_{1}\right)}^{2}+{\left({y}_{3}-{y}_{1}\right)}^{2}}$

Thus, we have found a formula for calculating the perimeter of the triangle whose vertices are given.

## Area and Perimeter of a Triangle - Key takeaways

- The perimeter of a triangle is the sum of all of its three sides.
- The formula for the perimeter of a triangle with sides $a,b\mathrm{and}c$
- The perimeter of a triangle whose vertices are $\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right)\mathrm{and}\left({x}_{3},{y}_{3}\right)$ is given by $P=\sqrt{{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}}+\sqrt{{\left({x}_{2}-{x}_{3}\right)}^{2}+{\left({y}_{2}-{y}_{3}\right)}^{2}}+\sqrt{{\left({x}_{3}-{x}_{1}\right)}^{2}+{\left({y}_{3}-{y}_{1}\right)}^{2}}$.
- The perimeter of a triangle for which the length of two sides are $a,b$
*c*is evaluated using the cosine rule: ${c}^{2}={a}^{2}+{b}^{2}-2ab\mathrm{cos}C$.

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##### Frequently Asked Questions about Perimeter of a Triangle

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

The formula of the perimeter of a triangle with sides *a, b, c *is *P = a + b + c.*

How do you find the missing perimeter of a triangle?

The missing perimeter is calculated by determining all the sides of the triangle and adding them.

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

For a triangle with two equal sides, *a *each and *b *being the third one, the perimeter is given by *P = 2a + b.*

How do you find the perimeter of a triangle with variables?

The perimeter is calculated the same way when the constants are given, instead the constants are replaced by variables but the formula remains the same.

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