Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It only applies to right triangles — triangles with one 90° angle — and it's one of the oldest results in mathematics still taught essentially unchanged, with evidence of the relationship appearing in Babylonian clay tablets more than a thousand years before Pythagoras, though his school gave it the first general proof that survives in the Western mathematical tradition.
Because it relates all three sides, it can be rearranged to find any one side once the other two are known: the hypotenuse directly by adding the squared legs and taking the square root, or a leg by subtracting the known leg's square from the hypotenuse's square before taking the square root.
Beyond triangle geometry, this same relationship underlies the distance formula for two points on a coordinate plane (the straight-line distance is literally the hypotenuse of a right triangle formed by the horizontal and vertical differences between the points), and it generalizes to non-right triangles as the Law of Cosines, which reduces exactly to a² + b² = c² when the included angle happens to be 90°.
What Each Variable Means
When to Use It
- Finding a missing side length in a right triangle
- Checking whether a triangle with three known side lengths actually has a right angle (if a² + b² = c², it does)
- As the basis for the distance formula between two coordinate points
Step-by-Step Example
Problem: A right triangle has legs of length 3 and 4. Find the hypotenuse.
a and b are the two shorter sides.
a = 3, b = 4Compute a² + b².
3² + 4² = 9 + 16 = 25c is the square root of that sum.
c = √25Interactive Calculator
Solving for Other Variables
a = √(c² − b²)Solve for one leg when the hypotenuse and the other leg are known.Common Mistakes
Mistake: Applying the theorem to a triangle that isn't a right triangle.
Fix: a² + b² = c² only holds when the triangle has a 90° angle — verify that first, or use the Law of Cosines for a general triangle.
Mistake: Mixing up which side is the hypotenuse.
Fix: The hypotenuse is always the longest side and always opposite the right angle — it's never one of the two legs being added.
Practice Questions
A right triangle has legs of length 6 and 8. Find the hypotenuse.
Hint: 6² + 8² = 36 + 64 = 100.
A ladder 13 m long leans against a wall, its base 5 m from the wall. How high up the wall does it reach?
Hint: Solve for the missing leg: √(13² − 5²).
Frequently Asked Questions
Does this work for any triangle?
No — only right triangles. For any other triangle, use the Law of Cosines, which reduces to the Pythagorean theorem when the included angle is 90°.
Can I use it to check if a triangle is a right triangle?
Yes — if the square of the longest side equals the sum of the squares of the other two, the triangle has a right angle opposite that longest side.
Related Formulas
Distance Formula
Calculates the straight-line distance between two points on a coordinate plane.
Learn more →Law of Cosines
A generalization of the Pythagorean theorem that works for any triangle, not just right triangles.
Learn more →