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# How to Find The Length Of A Triangle?

The length of a triangle refers to the summation of the three sides of a triangle. In other words, we can say, that the perimeter of A triangle is also known as the length of the triangle. While finding the length of a triangle, we will need the three sides of the triangle. Then we will add them together to get the perimeter of the triangle.

But that is not always given to us. Sometimes there is a situation where Three Side Lengths are Known. Sometimes When Two Sides are Known with one right angle. Sometimes we have situations like two side lengths, and the angle between them are known.

Let’s explore different situations and how to tackle them.

Situation 1:

Given Three Sides Of The Triangle –

This is the first and simplest case for finding the length or the perimeter of a triangle as we have already discussed that the perimeter or the length of a triangle is the addition of all the sides of the triangle. Now add all the sides of the triangle. The result will be the length of the triangle.

Situation 2:

Given Right Angle Triangle With Two Sides-

We all know that a Right Angle Triangle is a triangle which has one right angle. The longest side of the triangle, opposite to the right angle is known as the hypotenuse. While working with right angle triangles, we often came across the relations between the sides of the triangle. Fortunately, there is a universal formula that shows the relations between the sides of the triangle.

i.e. a2 + b2 = c2,

where a & b are the two sides adjacent to the right angle whereas c is the hypotenuse.

Let’s say we have a right angle triangle with known values of the two adjacent sides of the right angle which are 3cm and 4 cm respectively. As we have already identified the relation formula between the sides, let’s plug in the values in the equation.

Now, after plugging in we have,

32 + 42 = c2

=> c2 = 9 + 16

=> c2 = 25

=> c = 5

Hence, the length of the hypotenuse is 5 cm. Now that we have all sides with us, the perimeter of the triangle will be,

3 + 4 + 5 = 12cm

Hence, the length of the triangle is 12 cm.

Situation 3:

Given An SAS Triangle –

A SAS Triangle is the triangle whose two sides and the angle between are given. In this case, we will be using a special trigonometric formula to find the last side of the triangle. The formula is known as the law of Cosines. The formula is c2 = a2 + b2 – 2ab cos(C).

In this formula, the first side is denoted as “a” & the angle in front of it is denoted by “A”. In the same way, the side “b” faces the angle “B”, and the side “c” faces the angle “C”.

Let’s say we have

a = 10cm

b = 12cm

& C = 970

Now just input the values we have in the respective places. The final equation will be,

c2 = a2 + b2 – 2ab cos(C)

=> c2 = 102 + 122 – 2 X 10 X 12 cos(97)

=> c2 = 100 + 144 – 240 (-0.12187)

=> c2 = 244 – (-29.25)

=> c2 = 244 + 29.25

=> c2 = 273.25

=> c = 16.53

Hence the last side is 16.53 cm.

Now the length of the triangle is,

10 + 12 + 16.53

= 38.53 cm

Hence, the length of the triangle is 38.53 cm.