How to Find the Perimeter of a Triangle

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Finding the perimeter of a triangle means finding the distance around the triangle. [1] The simplest way to find the perimeter of a triangle is to add up the length of all of its sides, but if you don't know all of the side lengths you will need to calculate them first. This article will first teach you to find the perimeter of a triangle when you do know all three side lengths; this is the easiest and most common way. It will then teach you to find the perimeter of a right triangle when only two of the side lengths are known. Finally, it will teach you to find the perimeter of any triangle for which you know two side lengths and the angle measure between them (an "SAS Triangle"), using the Law of Cosines.

Method 1
Method 1 of 3:

Finding the Perimeter When Three Side Lengths are Known

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  1. For a triangle with sides a , b and c , the perimeter P is defined as: P = a + b + c . [2]
    • What this formula means in simpler terms is that to find the perimeter of a triangle, you just add together the lengths of each of its 3 sides.
  2. In this example, the length of side a = 5 , the length of side b = 5 , and the length of side c = 5 .
    • This particular example is called an equilateral triangle, because all three sides are of equal length. But remember that the perimeter formula is the same for any kind of triangle.
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  3. Add the three side lengths together to find the perimeter . In this example, 5 + 5 + 5 = 15 . Therefore, P = 15 .
    • In another example, where a = 4 , b = 3 , and c=5 , the perimeter would be: P = 3 + 4 + 5 , or 12 .
  4. If the sides of the triangle are measured in centimeters, then your answer should also be in centimeters. If the sides are measured in terms of a variable like x, your answer should also be in terms of x. [3]
    • In this example, the side lengths are each 5cm, so the correct value for the perimeter is 15cm.
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Method 2
Method 2 of 3:

Finding the Perimeter of a Right Triangle When Two Sides are Known

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  1. A right triangle is a triangle that has one right (90 degree) angle. The side of the triangle opposite the right angle is always the longest side, and it is called the hypotenuse. Right triangles show up frequently on math tests, and fortunately there is a very handy formula for finding the length of unknown sides!
  2. 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, a 2 + b 2 = c 2 . [4]
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    Joseph Meyer

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    Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.
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    Use this visual trick to understand the Pythagorean Theorem. Imagine a right triangle with squares constructed on each leg and the hypotenuse. by rearranging the smaller squares within the larger square, the areas of the smaller squares (a² and b²) will add up visually to the area of the larger square (c²).

  3. Remember that the longest side of the triangle is called the hypotenuse. It will be opposite the right angle and must be labeled c . Label the two shorter sides a and b . It doesn't really matter which is which, the math will turn out the same!
  4. Remember that a 2 + b 2 = c 2 . Substitute the side lengths in for the corresponding letters in the equation. [5]
    • If, for example, you know that side a = 3 and side b = 4 , then plug those values into the formula as follows: 3 2 + 4 2 = c 2 .
    • If you know the length of side a = 6 , and the hypotenuse c = 10 , then you should set the equation up like so: 6 2 + b 2 = 10 2 .
  5. You will first need to square the known side lengths which means multiplying each value by itself (for example 3 2 = 3 * 3 = 9). If you are looking for the hypotenuse, simply add the two values together and find the square root of this number to find the length. If it is a side length you are missing, you must do a bit of easy subtraction, and then take the square root to get your side length. [6]
    • In the first example, square the values in 3 2 + 4 2 = c 2 and find that 25= c 2 . Then calculate the square root of 25 to find that c = 5 .
    • In the second example, square the values in 6 2 + b 2 = 10 2 to find that 36 + b 2 = 100 . Subtract 36 from each side to find that b 2 = 64 , then take the square root of 64 to find that b = 8 .
  6. Recall that the perimeter P = a + b + c . Now that you know the lengths of sides a , b and c , you simply need to add the lengths together to find the perimeter. [7]
    • In our first example, P = 3 + 4 + 5, or 12 .
    • In our second example, P = 6 + 8 + 10, or 24 .

    Do you have the perimeter and are missing one side? Then you should subtract the sum of the two sides from the perimeter. This number equals the length of the missing side.

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Method 3
Method 3 of 3:

Finding the Perimeter of an SAS Triangle Using the Law of Cosines

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  1. Learn 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, and is a very useful formula. The Law of Cosines states that for any triangle with sides a , b , and c , with opposite angles A , B , and C : c 2 = a 2 + b 2 - 2ab cos (C) . [8] [9]
  2. The first side that you know should be labeled a , and the angle opposite it is A . The second side that you know should be labeled b ; the angle opposite it is B . The angle that you know should be labeled C , and the third side, the one you need to solve in order to find the perimeter of the triangle, is side c . [10]
    • For example, imagine a triangle with side lengths 10 and 12, and an angle between them of 97°. We will assign variables as follows: a = 10 , b = 12 , C = 97°.
  3. You will first need to find the squares of a and b, and add them together. Then find the cosine of C using the cos function on your calculator, or an online cosine calculator. [11] Multiply cos (C) by 2ab and subtract the product from the sum of a 2 + b 2 . The result is c 2 . Find the square root of this value and you have the length of side c . [12] Using our example triangle:
    • c 2 = 10 2 + 12 2 - 2 × 10 × 12 × cos (97) .
    • c 2 = 100 + 144 – (240 × -0.12187) (Round the cosine to 5 decimal places.)
    • c 2 = 244 – (-29.25)
    • c 2 = 244 + 29.25 (Carry the minus symbol through when cos (C) is negative!)
    • c 2 = 273.25
    • c = 16.53
  4. Recall that Perimeter P = a + b + c , so all you need to do is add the length you just calculated for side c to the values you already had for a and b .
    • In our example: 10 + 12 + 16.53 = 38.53 , the perimeter of our triangle!
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  • Question
    Can you find the perimeter if only one side is given?
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    This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.
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    If you know only 1 side but all 3 angles, you can use the rule of sines to find the remaining sides, then calculate the perimeter. If you know 1 side and 1 angle, you won’t be able to find the perimeter unless you’re dealing with a right triangle. For right triangles, you know that 1 angle is always 90°, so if you know another angle, you can use the sum of angles (180°) to figure out the third one. From there, you can use the laws of sine and cosine to figure out the other sides.
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  • Question
    How can you find the perimeter of a triangle if one side is missing?
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    This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.
    wikiHow Staff Editor
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    If it’s a right triangle, you can use the Pythagorean theorem (a2 + b2 = c2) to find the length of the missing side. From there, you can easily calculate the perimeter. For other types of triangles, you can use the law of cosines to find the perimeter if you know 2 sides and at least 1 of the angles.
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  • Question
    How can you find the missing side of a triangle given the perimeter?
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    This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.
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    Since the perimeter is the sum of the lengths of all sides, you can solve for the missing side by subtracting the lengths of the other 2 sides from the perimeter.
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      References

      1. ↑ http://www.mathsisfun.com/geometry/perimeter.html
      2. ↑ https://www.cuemath.com/measurement/perimeter-of-a-triangle/
      3. ↑ https://www.omnicalculator.com/math/triangle-perimeter
      4. ↑ https://www.varsitytutors.com/basic_geometry-help/how-to-find-the-perimeter-of-a-right-triangle
      5. ↑ https://sciencing.com/perimeter-right-triangle-6196682.html
      6. ↑ https://sciencing.com/perimeter-right-triangle-6196682.html
      7. ↑ https://www.cuemath.com/measurement/perimeter-of-right-angled-triangle/
      8. ↑ https://www.mathsisfun.com/algebra/trig-cosine-law.html
      9. ↑ https://www.cimt.org.uk/projects/mepres/step-up/sect4/index.htm
      1. ↑ https://www.calculatorsoup.com/calculators/geometry-plane/triangle-theorems.php
      2. ↑ http://www.rapidtables.com/calc/math/Cos_Calculator.htm
      3. ↑ https://www.mathsisfun.com/algebra/trig-solving-sas-triangles.html

      About This Article

      Article Summary X

      To find the perimeter of a triangle, use the formula perimeter = a + b + c, where a, b, and c are the lengths of the sides of the triangle. For example, if the length of each side of the triangle is 5, you would just add 5 + 5 + 5 and get 15. Therefore, the perimeter of the triangle is 15. If you only know the length of 2 of the triangle’s sides, you can still find the perimeter if it’s a right triangle, which means the triangle has one 90-degree angle. Just use the Pythagorean theorem, which is a^2+ b^2 = c^2, where a and b are the lengths of the known sides and c is the length of the unknown hypotenuse. For example, if the length of the known sides are 3 and 4, you would just add 3^2+ 4^2, or 9 + 16, and get 25. Then, you would take the square root of 25 to find c, which is 5. Therefore, the length of the unknown side is 5. Finally, add all of the side lengths together to find the perimeter. In this case you would add 3 + 4 + 5 and get 12. Therefore, the perimeter of the triangle is 12. If you want to learn how to solve the perimeter of your triangle if you only know 2 sides and an angle, keep reading the article!

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