Author:
Florence Bailey
Date Of Creation:
28 March 2021
Update Date:
1 July 2024
Content
- Steps
- Method 1 of 3: Three Sides
- Method 2 of 3: Along two sides of a right triangle
- Method 3 of 3: Along the two sides and the angle between them
The perimeter of a triangle is the total length of all its sides. The easiest way to find the perimeter of a triangle is to add the lengths of all its sides, but if you do not know the length of at least one side of the triangle, you must first find it. The first section of this article describes how to calculate the perimeter of a triangle from three known sides - this is the simplest and most common method. Then it is shown how to find the perimeter of a right triangle if the lengths of the two sides are known. And finally, it is described how, using the cosine theorem, to calculate the perimeter of any triangle, given two sides and the angle between them.
Steps
Method 1 of 3: Three Sides
1 Remember the formula for calculating the perimeter of a triangle. If the triangle has sides a, b and c, its perimeter P is equal to: P = a + b + c. - Thus, to find the perimeter of a triangle, add the lengths of all three of its sides.
2 Look at the triangle and find out the lengths of all three sides. Suppose a triangle has the following sides: a = 5, b = 5 and c = 5. - The triangle in question is called equilateral, since all three of its sides have the same length. However, the formula for calculating the perimeter is valid for any triangle.
3 Add the lengths of all three sides to find the perimeter. In our example 5 + 5 + 5 = 15, i.e P = 15. - Let's consider another example: a = 4, b = 3 and c = 5... In this case, the perimeter is: P = 3 + 4 + 5 = 12.
4 Do not forget to indicate the unit of measurement in your answer. If the sides are measured in centimeters, the final answer must also be given in centimeters. The answer must be in the same units as the lengths of the sides in the problem statement. - In the example shown, each side is 5 centimeters long, so the perimeter is 15 centimeters.
Method 2 of 3: Along two sides of a right triangle
1 Remember what a right triangle is. A rectangular triangle is such a triangle, one of the corners of which is right, that is, equal to 90 degrees. The longest side of such a triangle always lies opposite the right angle and is called the hypotenuse. The other two sides forming a right angle are called legs. Right-angled triangles are very common in math problems. Fortunately, there is a formula with which you can always calculate the length of the unknown side! 
2 Remember the Pythagorean theorem. This theorem states that in any right-angled triangle with legs a and b and hypotenuse c the sides are connected by the following relationship: a + b = c. 
3 Draw a right triangle and label the sides as a, b and c. The longest side of a right triangle is the hypotenuse. It lies opposite a right angle. Label the hypotenuse as cand the shorter sides are like a and b... It doesn't matter which leg you designate with a letter aand which one is a letter bas this will not affect the final result. 
4 Plug in the values of the known sides into the formula. remember, that a + b = c... Instead of letters, substitute the numbers given in the problem statement. - Suppose in the condition given that a = 3 and b = 4, then we get: 3 + 4 = c.
- If the leg a = 6 and hypotenuse c = 10, then you can write: 6 + b = 10.
5 Solve the resulting equation to find the unknown side. To do this, first square the known side lengths (just multiply this number by itself, for example 3 = 3 * 3 = 9). If you are looking for the hypotenuse, add the squares of the two sides and extract the square root from that sum. If you need to find a leg, subtract the square of the known leg from the square of the hypotenuse and extract the square root from the resulting number. - In the first example, add the squares of the sides 3 + 4 = c and we get 25 = c... After that, we extract the square root of 25 and find c = 5.
- In the second example, add the squares of the sides 6 + b = 10 and we get 36 + b = 100... Move 36 to the right side of the equation: b = 64... Take the square root of 64 and find b = 8.
6 Add the lengths of the three sides to find the perimeter. As we remember, the perimeter is calculated by the formula: P = a + b + c... After we have found the lengths of the sides a, b and c, you need to fold them to define the perimeter. - In the first example: P = 3 + 4 + 5 = 12.
- In the second example: P = 6 + 8 + 10 = 24.
Method 3 of 3: Along the two sides and the angle between them
1 Learn the cosine theorem. This theorem allows you to calculate the unknown side of a triangle if you are given the lengths of the other two sides and the angle between them. The cosine theorem is very useful, it is true for all triangles. This theorem states that for any triangle with sides a, b and c and opposite corners A, B and C the following formula is valid: c = a + b - 2ab cos(C). 
2 Give designations to the sides and corners of the triangle. Label the first known side as a, and the opposite angle is like A... Designate the second known side and the corner opposite to it, respectively. b and B... The known angle between these sides is designated as C, and the opposite side, the length of which must be found, as c. - Suppose you are given a triangle with sides 10 and 12 and an angle of 97 ° between them. In this case, we have: a = 10, b = 12, C = 97 °.
3 Plug the known values into the formula and find the unknown side with. First, square the lengths of the known sides and add the resulting values. Then find the cosine of angle C using a regular or online calculator. Multiply cos(C) on the 2ab and subtract the resulting number from the sum a + b... As a result, you will get c... Extract the square root to find the length of the unknown side c... In our example, we have: - c = 10 + 12 - 2 × 10 × 12 × cos(97°).
- c = 100 + 144 - (240 × -0.12187) (we have rounded up the cosine value to 5 decimal places).
- c = 244 - (-29.25).
- c = 244 + 29.25 (two minuses give a plus!).
- c = 273.25.
- c = 16.53.
4 Use the calculated side length cto find the perimeter of the triangle. Recall that the perimeter is calculated by the formula: P = a + b + c, that is, it should be added to the known values of the sides a and b found side length c. - In our example, we get: 10 + 12 + 16,53 = 38,53... So, the perimeter of the triangle is 38.53!
