The LessonThe length of the longest side of a right triangle (called the hypotenuse) can be found from Pythagoras' theorem, if the lengths of the other two sides are known. The hypotenuse is found using the formula:
In the formula, c is the length of the hypotenuse and a and b are the lengths of the other sides. The image below shows what we mean:
How to Find the Hypotenuse Using Pythagoras' Theorem
QuestionWhat is the hypotenuse of the right triangle below?
Find the lengths of the sides from the right triangle. In our example, the two shorter side lengths are a = 5 and b = 12.
Substitute a = 5 and b = 12 into the formula.
c = √(52 + 122) = √((5 × 5) + (12 × 12)) c = √(25 + 144) = √169 c = 13
Answer:The length of the hypotenuse is 13.
A Real Example of How to Find the Hypotenuse Using Pythagoras' TheoremIn the example above, we have used the formula c = √(a2 + b2) to find the length of the hypotenuse, c. This is just a rearrangement of the more memorable formula, a2 + b2 = c2 (see Note). If you find this simpler formula easier to remember, use it! Substitute in the lengths you know (replace the letters a and b with numbers) and then rearrange to find the hypotenuse, c.
Lesson SlidesThe slider below gives a real example of how to find the hypotenuse using Pythagoras' theorem. Open the slider in a new tab
Interactive WidgetHere is an interactive widget to help you learn about Pythagoras' theorem.
Rearranging Pythagoras' TheoremPythagoras' theorem states:
Taking square roots of both sides:
This gives the formula for finding the hypotenuse when the other sides are known:
Leaving the Answer as a Square RootThe square of the hypotenuse ,c2, will not always be a square number. When you square root c2, to find c, the answer will not be a whole number. It is sometimes best to leave the answer as a square root (also called a surd). For example, find the hypotenuse for the right triangle below, where a = 1 and b = 2:
a2 + b2 = c2 12 + 22 = c2 1 + 4 = c2 5 = c2Taking the square root gives c = 2.236, but it would be neater to write √5.