How to Find the Perimeter of a Triangle

Finding the perimeter of a triangle means finding the length around that triangle. ^{[1] X Research Source} The simplest way to find the perimeter of a triangle is to add up the lengths of all the sides. However, if you don’t know the lengths of the sides, you have to find them first. At the beginning, this article will show you how to find the perimeter of a triangle knowing the lengths of its three sides – that’s the easiest and most common way. Next, the article will teach you how to find the perimeter of a right triangle when only the lengths of two sides are known, and finally find the perimeter of any triangle knowing the lengths of the two sides and the angle between those two sides. (“Side-angle-triangle”) using the Cosine theorem.

Table of Contents

Find the perimeter when the lengths of the three sides of a triangle are known

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Recall the formula for the perimeter of a triangle. With a triangle with three sides a , b and c respectively , the perimeter P is determined as follows : P = a + b + c .

Simply put, to find the perimeter of a triangle, we simply add the lengths of their three sides.

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Looking at the triangle need to calculate the perimeter and determine the lengths of their three sides. In this example, the length of side a = 5 , the length of side b = 5 , and the length of side c = 5 .

The triangle in this particular example is called an equilateral triangle, because all three of its sides are of equal length. Remember, though, that the formula for perimeter is the same for all types of triangles.

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Add the lengths of the three sides to find the perimeter. In this example, 5 + 5 + 5 = 15 . Therefore, P = 15 .

In another example, for a = 4 , b = 3 , and c=5 , the perimeter of that triangle would be: P = 3 + 4 + 5 , which is 12 .

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Remember to include units in your final answer. If the sides of the triangle are measured in cm, then the answer is also in cm. If the edge is measured by a variable like x, your answer will also be expressed in terms of x.

In the above example, the length of each side is 5 cm, so the exact value of the circumference is 15 cm.

Find the perimeter of a right triangle when two of its sides are known

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Remember what a right triangle is. A right triangle is a triangle with a right angle (90 degrees). The side opposite the right angle is always the longest side of the triangle, called the hypotenuse. Right triangles come up frequently in math tests, and luckily, there’s a very handy formula for finding the lengths of unknown sides!

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Remember the Pythagorean theorem. The Pythagorean theorem says that for every right triangle with the lengths of two sides a and b respectively, and the length of the hypotenuse c, we have: a ² + b ² = c ² . ^{[2] X Research Source}

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Look at your triangle and name the sides “a”, “b”, and “c”. Remember that the longest side of a right triangle is called the hypotenuse. That is the side opposite the right angle and should be c . Name the two shorter sides a and b . It doesn’t matter which is a and which is b , because the result will be the same!

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Enter the side length you already know into the Pythagorean theorem. Remember that a ² + b ² = c ² . Substitute the lengths of the sides into the corresponding letter in the equation.

For example, if you know that side a = 3 and side b = 4 , then substitute these values into the formula like this: 3 ² + 4 ² = c ² .
If the length of the side a = 6 , and the hypotenuse c = 10 , then the equation would be set up as follows: 6 ² + b ² = 10 ² .

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Solve the equation, find the missing side length. First, you need to square the known side lengths, that is, multiply the length value by itself (eg 3 ² = 3 * 3 = 9). If you’re looking for the hypotenuse, just add the two values together and find the square root of the result. If it’s the length of the side adjacent to the right angle, you’ll have to do a simple subtraction, then take the square root to determine the length of the side to find.

In the first example, we squared the values in 3 ² + 4 ² = c ² to get 25= c ² . Next, take the square root of 25 to get c = 5 .
In the second example, we square the values in 6 ² + b ² = 10 ² to get 36 + b ² = 100 . Subtract both sides by 36 to get b ² = 64 , then take the square root of 64 to get b = 8 .

Find the perimeter of a side-angled triangle using the Cosine theorem

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Learn the Cosine theorem. The Cosine Theorem allows you to solve any triangle knowing the lengths of two sides and the measure of the angle between them. It applies to all triangles and is a very useful formula. The Cosine theorem says that, for every triangle with sides a , b , and c , and opposite angles respectively A , B , and C , we have: c ² = a ² + b ² – 2ab cos (C) ) . ^{[3] X Research Sources}^{[4] X Research Sources}

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Take a look at your triangle and assign letters representing variables to its components. The first side you know should be a , and the opposite angle is angle A . The second edge you know should be b ; The angle opposite it is angle B. The angle you know will probably be angle C , and the third side, the side to find to calculate the perimeter of the triangle is c .

For example, imagine a triangle with sides 10 and 12 respectively, and the angle between them measures 97°. We will assign variables like this: a = 10 , b = 12 , C = 97°.

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Substitute your information into the equation and solve the equation to find the side c. You will have to find the squares of a and b first, then add them together. Next, find the cosine of C using the cosine function on your pocket calculator or online cosine calculator. ^{[5] X Research Source} Multiply cos (C) by 2ab and subtract that product from the sum of a ² + b ² . The result obtained is c ² . Find the square root of this value and you have the length of side c . With the triangle in the example above:

c ² = 10 ² + 12 ² – 2 × 10 × 12 × cos (97) .
c ² = 100 + 144 – (240 × -0.12187) (Round cosine to five decimal places).
c ² = 244 – (-29.25)
c ² = 244 + 29.25 (Bring a minus sign because cos (C) is negative!)
c ² = 273.25
c = 16.53

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Use side length c to calculate the perimeter of the triangle. Remember that the perimeter P = a + b + c , so you just add the length you just calculated for side c with the existing values of a and b .

In this example: 10 + 12 + 16.53 = 38.53 , which is the perimeter of the triangle to find!

Steps

Find the perimeter when the lengths of the three sides of a triangle are known

Find the perimeter of a right triangle when two of its sides are known

Find the perimeter of a side-angled triangle using the Cosine theorem

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