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How To Find the Third Side of a Triangle in 3 Ways

There are several different ways you can compute the length of the third side of a triangle. Depending on whether you need to know how to find the third side of a triangle on an isosceles triangle or a right triangle, or if you have two sides or two known angles, this article will review the formulas that you need to know.

Pythagorean Theorem for the Third Side of a Right Angle Triangle

The Pythagorean Theorem is used for finding the length of the hypotenuse of a right triangle.

Pythagoras was a Greek mathematician who discovered that on a triangle abc, with side c being the hypotenuse of a right triangle (the opposite side to the right angle), that:

how to find the third side of a triangle: formula of side c being the hypotenuse of a right triangle

So, as long as you are given two lengths, you can use algebra and square roots to find the length of the missing side.

Formula for the Base of an Isosceles Triangle

If you know the side length and height of an isosceles triangle, you can find the base of the triangle using this formula:

how to find the third side of a triangle: formula for the base of an isosceles triangle

where a is the length of one of the two known, equivalent sides of the isosceles.

Find the Third Side of Any Triangle

Now that we've reviewed the two basic cases, let’s look at how to find the third unknown side for any triangle.

There are two additional concepts that you must be familiar with in trigonometry: the law of cosines and the law of sines. Both of them allow you to find the third length of a triangle.

The law of sines is the simpler one. It states that the ratio between the length of a side and its opposite angle is the same for all sides of a triangle:

how to find the third side of a triangle: formula of the law of sines

Here, A, B, and C are angles, and the lengths of the sides are a, b, and c.

how to find the third side of a triangle: graph of the law of sines

Because we know angle A and side a, we can use that to find side c.

how to find the third side of a triangle: formula on how to find side c

how to find the third side of a triangle: formula on how to find side c

how to find the third side of a triangle: formula on how to find side c

The length of side c is 4.38.

The law of cosines is slightly longer and looks similar to the Pythagorean Theorem. It states that:

formula of the law of cosines

Here, angle C is the third angle opposite to the third side you are trying to find.

how to find the third side of a triangle: graph of the law of cosines

Because we know the lengths of side a and side b, as well as angle C, we can determine the missing third side:

formula of how to determine the missing third side

formula of how to determine the missing third side

formula of how to determine the missing third side

formula of how to determine the missing third side

formula of how to determine the missing third side

formula of how to determine the missing third side

formula of how to determine the missing third side

Different Ways to Find the Third Side of a Triangle

There are a few answers to how to find the length of the third side of a triangle. To choose a formula, first assess the triangle type and any known sides or angles.

For a right triangle, use the Pythagorean Theorem. For an isosceles triangle, use the area formula for an isosceles. If you know some of the angles and other side lengths, use the law of cosines or the law of sines.

You’ll be on your way to knowing the third side in no time.

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