The Pythagorean Theorem
The relationship between the three sides of a right triangle.
The idea
In any right triangle, if a and b are the two legs and c is the hypotenuse (the side opposite the right angle, and always the longest side), then:
You can use this to find a missing side when you know the other two, or to check whether a triangle is a right triangle at all — if the sides satisfy a² + b² = c², the angle opposite the longest side is 90°.
Practice problems
A right triangle has legs of length 6 and 8. Find the hypotenuse.
Show answer
c = √(6² + 8²) = √(36 + 64) = √100 = 10.
A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.
Show answer
b = √(13² − 5²) = √(169 − 25) = √144 = 12.
A 10-foot ladder leans against a wall, with its base 6 feet from the wall. How high up the wall does it reach?
Show answer
height = √(10² − 6²) = √(100 − 36) = √64 = 8 feet.
Is a triangle with sides 7, 24, and 25 a right triangle?
Show answer
7² + 24² = 49 + 576 = 625, and 25² = 625. Since they're equal, yes, it's a right triangle.