Find the Perimeter of a Triangle

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.

Steps

Finding the Perimeter When Three Side Lengths are Known

  1. Remember the formula for finding the perimeter of a triangle. For a triangle with sides a, b and c, the perimeter P is defined as: P = a + b + c.
    • 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. Look at your triangle and determine the lengths of the three sides. 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.
  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. Remember to include the units in your final answer. 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.
    • In this example, the side lengths are each 5cm, so the correct value for the perimeter is 15cm.

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

  1. Remember what a right triangle is. 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, a2 + b2 = c2.[2]
  3. Look at your triangle, and label the sides "a," "b," and "c". 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. Enter the side lengths that you know into the Pythagorean Theorem. Remember that a2 + b2 = c2. Substitute the side lengths in for the corresponding letters in the equation.
    • If, for example, you know that side a = 3 and side b = 4, then plug those values into the formula as follows: 32 + 42 = c2.
    • If you know the length of side a = 6, and the hypotenuse c = 10, then you should set the equation up like so: 62 + b2 = 102.
  5. Solve the equation to find the missing side length. You will first need to square the known side lengths which means multiplying each value by itself (for example 32 = 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.
    • In the first example, square the values in 32 + 42 = c2 and find that 25= c2. Then calculate the square root of 25 to find that c = 5.
    • In the second example, square the values in 62 + b2 = 102 to find that 36 + b2 = 100. Subtract 36 from each side to find that b2 = 64, then take the square root of 64 to find that b = 8.
  6. Add up the lengths of the three side lengths to find the perimeter. 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.
    • In our first example,P = 3 + 4 + 5, or 12.
    • In our second example, P = 6 + 8 + 10, or 24.

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

  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: c2 = a2 + b2 - 2ab cos(C).[3][4]
  2. Look at your triangle and assign variable letters to its components. 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.
    • 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. Plug your information into the equation and solve for side c. 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.[5] Multiply cos(C) by 2ab and subtract the product from the sum of a2 + b2. The result is c2. Find the square root of this value and you have the length of side c.
    Using our example triangle:
    • c2 = 102 + 122 - 2 × 10 × 12 × cos(97).
    • c2 = 100 + 144 – (240 × -0.12187) (Round the cosine to 5 decimal places.)
    • c2 = 244 – (-29.25)
    • c2 = 244 + 29.25 (Carry the minus symbol through when cos(C) is negative!)
    • c2 = 273.25
    • c = 16.53
  4. Use side length c to find the perimeter of the triangle. 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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Sources and Citations

  1. ↑ http://www.mathsisfun.com/geometry/perimeter.html
  2. ↑ https://www.wikihow.com/Use-the-Pythagorean-Theorem
  3. ↑ https://www.mathsisfun.com/algebra/trig-cosine-law.html
  4. ↑ http://www.cimt.plymouth.ac.uk/projects/mepres/step-up/sect4/index.htm
  5. ↑ http://www.rapidtables.com/calc/math/Cos_Calculator.htm
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