The formula a2 + b2 = c2 connects the three sides of every right triangle. This guide works through the proof, two complete examples, and the places you actually meet this theorem outside a classroom.
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.
That single sentence has been doing heavy lifting for roughly 2,500 years. Surveyors used it before Pythagoras got credit for it. Carpenters still use it every day. And it shows up quietly in everything from GPS coordinates to video game collision detection.
For a right triangle with legs a and b and hypotenuse c:
The legs are the two shorter sides that form the right angle. The hypotenuse is always the side opposite that right angle, and it is always the longest of the three. If you forget which side is which, look for the 90-degree corner and the hypotenuse is the side that does not touch it.
Rearranging gives you two more useful forms. To find a missing leg, subtract the known squares:
And to isolate the hypotenuse on its own:
Suppose a right triangle has legs a = 3 and b = 4. What is c?
So the three sides are 3, 4, and 5. Every measurement here is a whole number, which makes the 3-4-5 triangle the most recognized example in the subject. You can run the same calculation faster with the Pythagorean theorem calculator.
Now reverse the problem. The hypotenuse is c = 13 and one leg is b = 5. Find a.
The sides 5, 12, and 13 satisfy the equation exactly. This is another clean whole-number solution, and it comes up often enough that it is worth memorizing alongside 3-4-5.
There are more than 370 known proofs of this theorem. The most intuitive one uses area.
Draw a square with side length (a + b). Inside it, arrange four identical right triangles with legs a and b, each placed corner to corner so their hypotenuses form a smaller square in the center. The area of the big square is (a + b)2. That same area also equals the four triangles plus the inner square, which gives 4 times (1/2 ab) plus c2. Expand (a + b)2 to a2 + 2ab + b2, cancel the 2ab from both sides, and you are left with a2 + b2 = c2. No algebra tricks, just area accounting.
Khan Academy walks through this area proof with interactive diagrams if you want to see it step by step.
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy the equation exactly. They are useful because they let you work with right triangles without reaching for a calculator.
The three triples worth knowing first:
Multiplying all three numbers in any triple by the same whole number produces another valid triple. So 3-4-5 scaled by 3 gives 9-12-15, which still satisfies a2 + b2 = c2. Wolfram MathWorld maintains a thorough treatment of Pythagorean triples if you want to go further into the number theory.
Carpenters and builders use the 3-4-5 triple constantly to check whether a corner is truly square. The method is simple. Measure 3 feet along one wall from the corner and mark it. Measure 4 feet along the adjacent wall and mark that too. If the diagonal between those two marks is exactly 5 feet, the corner is at 90 degrees. If not, adjust until it is.
The same check works at any scale. A deck layout might use 6-8-10 feet (the 3-4-5 triple doubled) to get a longer baseline and reduce measurement error. The underlying math is identical. This is why the theorem matters to anyone who builds things, not just people who took geometry.
For problems that involve more than simple distances, the same core ideas show up in statistics as well. The formula for calculating spread in a data set, for instance, involves squaring differences and taking a root, which is structurally similar to what happens here. If you are working through quantitative problems, the standard deviation calculator handles that side of things.
The theorem is specific to right triangles. Acute and obtuse triangles do not follow a2 + b2 = c2. For an acute triangle the sum of the two smaller squares exceeds the largest square. For an obtuse triangle it falls short. The law of cosines generalizes the relationship to cover all triangle types, but that is a separate tool.
You can also test whether a triangle is a right triangle by checking the equation in reverse. If 62 + 82 = 102 (36 + 64 = 100), then the triangle with those three sides has a right angle. If the numbers do not balance, it does not.
The Pythagorean theorem calculator finds any missing side in seconds. Enter the two values you know and it returns the third with the steps shown.
In any right triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides. Written as a formula: a2 + b2 = c2, where c is the hypotenuse.
Square both legs, add the results, then take the square root. If a = 3 and b = 4, then c = sqrt(32 + 42) = sqrt(9 + 16) = sqrt(25) = 5.
No. It applies only to right triangles, meaning triangles with one 90-degree angle. For other triangles, the law of cosines generalizes the relationship.
A Pythagorean triple is a set of three positive whole numbers (a, b, c) that satisfy a2 + b2 = c2. Common examples are (3, 4, 5), (5, 12, 13), and (8, 15, 17).
Editor at Encore Editorial, Chris Terry sets the editorial standards here and turns dense topics into plain English. He has written widely on education, finance, and consumer markets.