In the left triangle, the measure of the hypotenuse is missing. Use the Pythagorean theorem to solve for the missing length.
Replace the variables in the theorem with the values of the known sides.
482 + 142 = c2
Square the measures and add them together.
The triangle on the right is missing the bottom length, but you do have the length of the hypotenuse. It doesn't matter whether you call the missing length a or b.
You’ll be asked how to find the length of a triangle over and over again in math and trigonometry. Maybe you need to find the missing side of a right triangle, maybe you know both side b and side c, or maybe you know only the opposite angle of the length of a side you are trying to find. In any case, we have formulas to help.
Case #1: When You Know the Area of a Triangle
If you know the area of a triangle and either the base or height, you can easily find the length by using the area formula:
Let’s use the formula to find the base of a triangle with an area of 20 and a height of 5:
This works for equilateral triangles and isosceles triangles as well!
Case #2: When You’re Finding the Length of a Right Triangle
To find the hypotenuse of a right triangle, use the Pythagorean Theorem. Start with the two known sides and use the famous formula developed by the Greek mathematician Pythagoras, which states that the sum of the squares of the sides is equal to the square of the length of the third side:
As an example, finding the length of the third side for a triangle with two other sides length 5 and 12:
From there you square the sides of the triangle, add them together, and compare them to the square root (sometimes abbreviated as sqrt) of the unknown side. Best of all? This works for all triangles that have a right angle. Just don't forget that c always refers to the hypotenuse or longest side of the triangle.
Case #3: When You’re Using the Law of Sines and the Law of Cosines
The Law of Sines says that for all angles of a triangle, the ratio of the sine of that angle to its opposite side will always be the same.
Here’s an example of the Law of Sines in action:
The length of side c is 2.98.
The Law of Cosines says you can determine the length of any triangle side if you know its opposite angle and the lengths of the other two sides.
Here’s an example of the Law of Cosines in action:
The Best Formula for Finding the Length of a Triangle
It all comes down to what information you start with. You’ll often know one or two sides of a triangle, missing angles, or other clues.
Review your formulas like the area formula, Pythagorean Theorem, and the Law of Sines, and the Law of Cosines, and you will be well equipped to find the length of any triangle!