# How to Find the Perimeter of a Triangle

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Finding the distance around the triangle is finding the perimeter of a triangle. Here is a step by step guide on how to find the perimeter of a triangle with three different methods.

There are three primary methods used to find the perimeter of a right triangle.

1. When side lengths are given, add them together.
2. Solve for a missing side using the Pythagorean theorem.
3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

## How to Find the Perimeter of a Triangle When Three Side Lengths are Known

1. Remember and write down the formula for finding the perimeter of a triangle.

For a triangle with sides ab and c, the perimeter P is defined as

### $\dpi{150}&space;P&space;=&space;a&space;+&space;b&space;+&space;c$$\dpi{150}&space;P&space;=&space;a&space;+&space;b&space;+&space;c$

2. Look at your triangle and determine the lengths of the three sides.

For Example: Let three sides of a traingle is a = 5 cm, b = 4 cm, c = 2 cm

3. Add the three side lengths together to find the perimeter.

$\dpi{150}&space;P&space;=&space;a&space;+&space;b&space;+&space;c&space;=&space;5&space;+&space;4&space;+&space;2&space;=&space;11$

4. Make sure to add units to your final answer.

Therefore, the answer is 11 cm.

## How to Find the Perimeter of a Right Triangle When Two Sides are Known

A right triangle is a triangle that has one right (90 degrees) angle. The side of the triangle opposite the right angle is always the longest side, and it is called the hypotenuse.

1. Recall the Pythagorean Theorem. The Pythagorean Theorem tells us that for any right triangle with sides of length a and b, and hypotenuse of length c, a2 + b2 = c2

2. Label the sides “a,” “b,” and “c” and the longest side of the triangle will be opposite the right angle and must be labeled c.

3. Enter the side lengths that you know into the Pythagorean Theorem.
For example, you know that side a = 3 and side b = 4, then plug those values into the formula as follows: 32 + 42 = c2.

4. Solve the equation to find the missing side length.

$\\a^{2}&space;+&space;b^{2}&space;=&space;c^{2}&space;\\3^{2}&space;+&space;4^{2}&space;=&space;c^{2}&space;\\9&space;+&space;16&space;=&space;c^{2}&space;\\c^{2}&space;=&space;25&space;\\&space;c&space;=&space;5$

5. Add up the lengths of the three side lengths to find the perimeter.

P = a + b + c
P = 3 + 4 + 5
P = 12

## How to Find the Perimeter of a Side-Angle-Side Triangle Using the Law of Cosines

The Law of Cosines allows you to solve any triangle when you know two side lengths and measurement of the angle between them. It works on any triangle. The Law of Cosines states that for any triangle with sides ab, and c, with opposite angles AB, and Cc2 = a2 + b2 – 2ab cos(C).

1. Look at your triangle and assign variable letters to its components.

For example, a triangle with side lengths 12 and 14, and an angle between them of 110°. We will assign variables as follows: a = 12, b = 14, C = 110°

2. Plug your information into the equation and solve for side c.

$\\C^{2}&space;=&space;a^{2}&space;+&space;b^{2}&space;-&space;2abcosC&space;\\C^{2}&space;=&space;12^{2}&space;+&space;14^{2}&space;-&space;2(12)(14)cos(110)&space;\\C^{2}&space;=&space;144&space;+&space;196&space;-&space;336(-0.34202)&space;\\C^{2}&space;=&space;340&space;-&space;(-114.91)&space;\\C^{2}&space;=&space;340&space;+&space;114.91&space;\\C^{2}&space;=&space;454.91&space;\\C\&space;=&space;21.32$

3. Use side length c to find the perimeter of the triangle.

P = a + b + c
P = 12 + 14 + 21.32
P = 59.32