3 Ways to Calculate the Perimeter of a Triangle

2024 Author: Samantha Chapman | [email protected]. Last modified: 2024-01-15 13:47

Finding the perimeter of a triangle means finding the measure of its outline. The simplest way to calculate it is to add the lengths of the sides together. However, if you don't know all of these values, you need to figure them out first. This article will teach you, first, to find the perimeter of a triangle by knowing the length of all three sides, then to calculate the perimeter of a right triangle of which you only know the measurements of two sides, and finally to deduce the perimeter. of any triangle of which you know the length of two sides and the width of the angle between them. In the latter case you will apply the Cosine Theorem.

Steps

Method 1 of 3: With Three Known Sides

Step 1. Remember the formula for the perimeter of a triangle

Considered a triangle of sides to, b And c, the perimeter P. is defined as: P = a + b + c.

In practice, to find the perimeter of a triangle you have to add the lengths of the three sides

Step 2. Check the problem figure and determine the value of the sides

For example, the side to =

Step 5., the side b

Step 5. and finally c

Step 5

This specific case concerns an equilateral triangle because the sides are equal to each other. But remember that the perimeter formula applies to any triangle

Step 3. Add the side values together

In our example: 5 + 5 + 5 = 15. Therefore P = 15.

• If we consider a = 4, b = 3 And c = 5, then the perimeter will be: P = 3 + 4 + 5 that is

  Step 12..

Step 4. Remember to indicate the unit of measurement

If the sides were measured in centimeters, the perimeter will also be expressed in centimeters. If the sides are expressed in the form of an “x” variable, the perimeter will be too.

In our initial example the sides of the triangle measure 5 cm each, so the perimeter is equal to 15 cm

Method 2 of 3: With Two Known Sides

Step 1. Remember the definition of a right triangle

A triangle is right when one of its angles is right (90 °). The side opposite the right angle is the longest and is called the hypotenuse. This type of triangle often appears in exams and class assignments but, luckily, there is a very simple formula to help you!

Step 2. Review the Pythagorean Theorem

His statement reminds us that in every right triangle with legs of length "a" and "b" and the hypotenuse of length "c": to² + b² = c².

Step 3. Check the triangle that is your problem and name the sides "a", "b" and "c"

Remember that the larger side is called the hypotenuse, it is opposite the right angle and must be indicated with c. Call the other two sides (the catheti) to And b. In this case it is not necessary to respect any order.

Step 4. Enter the known values in the Pythagorean Theorem formula

Remember that: to² + b² = c². Substitute the lengths of the sides for "a" and "b".

• If, for example, you know that a = 3 And b = 4, then the formula becomes: 3² + 4² = c².
• If you know that a = 6 and that the hypotenuse is c = 10, then the equation will be: 6² + b² = 10².

Step 5. Solve the equation to find the missing side

You must first raise the known values to the second power, i.e. multiply them by themselves (for example: 3² = 3 * 3 = 9). If you are looking for the value of the hypotenuse, simply add the squares of the legs together and then calculate the square root of the result you get. If you have to find the value of a cathetus, then you have to proceed with a subtraction and then extract the square root

• If we consider our first example: 3² + 4² = c², so 25 = c². We now calculate the square root of 25 and find that c = 5.
• In our second example, however: 6² + b² = 10² and we get that 36 + b² = 100. We subtract 36 from each side of the equation and we have: b² = 64, we extract the root of 64 to have b = 8.

Step 6. Add the sides together to find the perimeter

Remember that the formula is: P = a + b + c. Now that you know the values of to, b And c you can proceed to the final calculation.

• For the first example: P = 3 + 4 + 5 = 12.
• In the second example: P = 6 + 8 + 10 = 24.

Method 3 of 3: Using the Cosine Theorem

Step 1. Learn the Cosines Theorem

This allows you to solve any triangle for which you know the length of two sides and the width of the angle between them. It applies to any type of triangle and is a very useful formula. The Cosines Theorem states that for any triangle of sides to, b And c, with opposite sides TO, B. And C.: c² = a² + b² - 2ab cos (C).

Step 2. Look at the triangle you are looking at and assign the corresponding letters to each side

The first known side is named to and its opposite corner: TO. The second known side is called b and its opposite corner: B.. The known angle between "a" and "b" is said C. and the side opposite it (unknown) is indicated with c.

• Let's imagine a triangle with sides 10 and 12 enclosing an angle of 97 °. The variables are assigned as follows: a = 10, b = 12, C = 97 °.

  

  Step 3. Insert the known values into the Cosine Theorem formula and solve it for "c"

  First find the squares of "a" and "b" and then add them together. Calculate the cosine of C using the calculator's cos function or an online calculator. Multiply cos (C) for 2ab and subtract this product from the sum of to² + b². The result is equal to c². Take the square root of this result and you will get the side c. Let's proceed with the example above:

  • c² = 10² + 12² - 2 × 10 × 12 × cos (97).
  • c² = 100 + 144 – (240 × -0, 12187) (rounds the cosine value to the fifth decimal place).
  • c² = 244 – (-29, 25).
  • c² = 244 + 29, 25 (remove the minus sign from the brackets when cos (C) is a negative value!)
  • c² = 273, 25.
  • c = 16.53.

  

  Step 4. Use the length of the value of c to find the perimeter of the triangle

  Remember that P = a + b + c, so you just have to add to to And b you already notice the just calculated value of c.

  Always following our example: P = 10 + 12 + 16.53 = 38.53.