3 Ways to Find the Perimeter of a Triangle

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3 Ways to Find the Perimeter of a Triangle
3 Ways to Find the Perimeter of a Triangle
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Finding the perimeter of a triangle consists of finding the distance of the line that passes through its edges. The simplest way to do this is to add up the length of all sides, but if you don't know them yet, you need to calculate them first. This article will first teach you how to find the perimeter of a triangle when all three side lengths are known; this is the simplest and most common way. It will then teach you how to find the perimeter of a right triangle when only two of the side lengths are known. Finally, we'll teach you how to find the perimeter of any triangle of which you know two sides and the angle between them (a “CAC triangle”), with the Law of Cosines.

Steps

Method 1 of 3: Finding the Perimeter When Three Sides Are Known

Find the Perimeter of a Triangle Step 1

Step 1. Recall the formula for finding the perimeter of a triangle

For given triangle with sides The, B and ç, the perimeter FOR is defined as: P = a + b + c.

What this formula means, in simple terms, is that to find the perimeter of a triangle you only need to join the lengths of each of its three sides

Find the Perimeter of a Triangle Step 2

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

In this example, the length of the side a = 5, the one on the side b = 5 and the one on the side c = 5.

This particular example is called an equilateral triangle because all three sides have equal measurements. Remember, however, that the formula for the perimeter is the same for any type of triangle

Find the Perimeter of a Triangle Step 3

Step 3. Add up the lengths of the three sides to find the perimeter

In the present example, 5 + 5 + 5 = 15. Soon, P = 15.

  • In another example, in which a = 4, b = 3 and c = 5, the perimeter would be: P = 3 + 4 + 5, or

    Step 12..

Find the Perimeter of a Triangle Step 4

Step 4. Remember to include units in your final answer

If the sides of the triangle are measured in centimeters, the answer must also be given in centimeters. If they are given in terms of a variable like x, your answer must also be defined in terms of x.

In this example, the sides have a measurement of 5 cm, so the correct value for the perimeter is 15 cm

Method 2 of 3: Finding the Perimeter of a Right Triangle When Two Sides Are Known

Find the Perimeter of a Triangle Step 5

Step 1. Recall what a right triangle is

A right triangle is one that has a right angle (90 degrees). The side of the triangle opposite the right angle will always be the largest, being called the hypotenuse. Right triangles often appear on math tests, and luckily, there's a very useful formula for figuring out the value of unknown sides!

Find the Perimeter of a Triangle Step 6

Step 2. Recall the Pythagorean Theorem

The Pythagorean Theorem tells us that for every right triangle with sides of size a and b, and hypotenuse of size c, The2 + b2 = c2.

Find the Perimeter of a Triangle Step 7

Step 3. Look at your triangle and label sides “a”, “b” and “c”

Remember that the longest side is called the hypotenuse. It will be opposite the right angle and should be named ç. Name the two smallest sides as The and B. It doesn't really matter which one is represented by which letter - the result will be the same!

Find the Perimeter of a Triangle Step 8

Step 4. Enter the known side lengths in the Pythagorean Theorem

remember that The2 + b2 = c2. Replace the side lengths with the corresponding letters in the equation.

  • If, for example, you know that the side a = 3 and that side b = 4, enter these values ​​into the formula as follows: 32 + 42 = c2.
  • If you know the lengths of one side a = 6 and the hypotenuse c = 10, you need to describe the equation as follows: 62 + b2 = 102.
Find the Perimeter of a Triangle Step 9

Step 5. Solve the equation to find the length of the unknown side

You must first square the known side lengths, that is, multiply each value by itself (for example: 32 = 3 × 3 = 9). If you're looking for the hypotenuse, simply add the two values ​​together and find the square root of that number to find the length. If it's an unknown side length, you'll need to do some simple subtractions and then extract the square root to get the desired side length.

  • In the first example, square the values ​​present in 32 + 42 = c2 and find out that 25 = c2. Then calculate the square root of 25 to find that c = 25.
  • 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 and then extract the square root of 64 to get the result b = 8.
Find the Perimeter of a Triangle Step 10

Step 6. Add up the lengths of the three sides to find the perimeter

Remember the perimeter formula P = a + b + c. Now, knowing the value of the sides The, B and ç, you simply add up the lengths and figure out the perimeter.

  • In our first example, P = 3 + 4 + 5 = 12.
  • In our second example, P = 6 + 8 + 10 = 24.

Method 3 of 3: Finding the Perimeter of a CAC Triangle Using Cosine Law

Find the Perimeter of a Triangle Step 11

Step 1. Learn the Cosine Law

The Cosine Law allows you to unravel any triangle if you know the lengths of two sides and the measure of the angle between them. It works on any triangle and is a very useful formula. The Cosine Law states that for any triangle with sides The, B and ç, with opposite angles THE, B and Ç: ç2 = the2 + b2 - 2b cos (C).

Find the Perimeter of a Triangle Step 12

Step 2. Look at your triangle and assign variable letters to its components

The first known side should be called the The and the angle opposite to it, of THE. The second known side must be named B; the opposite angle to it, B. The known angle must be defined by Ç, and the third side, for which the problem must be solved in order to find the perimeter of the triangle, will be the ç.

  • For example, imagine a triangle with side lengths equal to 10 and 12, and an angle between them of 97°. We will define the variables as follows: a = 10, b = 12 and C = 97°.
Find the Perimeter of a Triangle Step 13

Step 3. Enter the known information into the equation and solve the problem to find side c

You must first find the squares of a and b, adding them up next. Then find the cosine of C with the cos function on your calculator or on an online cosine calculator. Multiply cos (C) per 2b and subtract the product from the sum of The2 + b2. The result will be equal to ç2. Find the square root of this value, and you get the size of the side ç. Using our triangle as an example:

  • ç2 = 102 + 122 - 2 × 10 × 12 × cos (97)
  • ç2 = 100 + 144 - (240 × -0, 12187)

    Round the cosine to 5 places

  • ç2 = 244 - (-29, 25)
  • ç2 = 244 + 29, 25

    When cos (C) is negative, remember the sign

  • ç2 = 273, 25
  • c = 16.53
Find the Perimeter of a Triangle Step 14

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

Remember that the perimeter P = a + b + c, so all that has to be done is to add the newly calculated length for the side ç to the values ​​already known for The and B. Easy!

  • In our example: 10 + 12 + 16, 53 = 38, 53, the perimeter of our triangle!

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