The Pythagorean theorem lets you find the missing side of any right triangle using one formula: a² + b² = c², where c is the longest side (the hypotenuse) and a and b are the two shorter sides. If you know two sides, you can always solve for the third. Here’s exactly how to do it.
The Formula and When It Works
The theorem applies only to right triangles, meaning one of the three angles is exactly 90 degrees. The side directly opposite that right angle is always the longest and is called the hypotenuse. The other two sides are called legs.
The relationship is simple: if you square the lengths of both legs and add them together, the result equals the square of the hypotenuse. Written out:
a² + b² = c²
This works every time, no matter the size of the triangle, as long as that 90-degree angle is present. It does not work on triangles without a right angle.
Finding the Hypotenuse
This is the most common use. You know both legs and need the longest side. Walk through it with real numbers: say one leg is 3 and the other is 4.
- Plug into the formula: 3² + 4² = c²
- Square each leg: 9 + 16 = c²
- Add: 25 = c²
- Take the square root: c = √25 = 5
The hypotenuse is 5. Another common example: legs of 5 and 12. Squaring gives you 25 + 144 = 169, and √169 = 13. So the hypotenuse is 13.
The key step people forget is that last one. After adding the squared legs together, you have c squared, not c itself. You must take the square root to get the actual side length. Stopping one step early is one of the most common mistakes students make.
Finding a Missing Leg
Sometimes you know the hypotenuse and one leg, and need the other leg. The algebra just rearranges slightly. If you know c and a, solve for b:
b² = c² − a²
For example, if the hypotenuse is 10 and one leg is 6:
- Plug in: b² = 10² − 6²
- Square: b² = 100 − 36
- Subtract: b² = 64
- Square root: b = √64 = 8
The missing leg is 8. Notice the critical difference here: when solving for a leg, you subtract. When solving for the hypotenuse, you add. Mixing these up is another frequent error. A simple way to keep it straight: the hypotenuse is always the largest number. If your answer comes out larger than the hypotenuse, something went wrong.
Pythagorean Triples Worth Memorizing
Certain sets of whole numbers satisfy the theorem perfectly, so recognizing them saves time. The most common ones are:
- 3, 4, 5 (the classic: 9 + 16 = 25)
- 5, 12, 13 (25 + 144 = 169)
- 8, 15, 17 (64 + 225 = 289)
- 7, 24, 25 (49 + 576 = 625)
Any multiple of these triples also works. Double the 3-4-5 triple and you get 6-8-10. Triple it and you get 9-12-15. If you spot one of these patterns in a problem, you can skip the calculation entirely.
Testing Whether a Triangle Is a Right Triangle
The theorem also works in reverse. If someone gives you three side lengths and asks whether the triangle is a right triangle, plug the two shorter sides into a² + b² and see if the result equals the square of the longest side.
Take a triangle with sides 5, 6, and 8. The longest side is 8, so check: 5² + 6² = 25 + 36 = 61. Then 8² = 64. Since 61 does not equal 64, this is not a right triangle.
Now try sides 6, 8, and 10. Check: 6² + 8² = 36 + 64 = 100. And 10² = 100. The equation holds, so this is a right triangle. This reverse check is called the converse of the Pythagorean theorem, and it works even when you don’t know any of the triangle’s angles.
Common Mistakes to Avoid
The biggest errors come down to three things. First, applying the theorem to a triangle that isn’t a right triangle. If there’s no 90-degree angle, this formula doesn’t apply. Second, mixing up which side is the hypotenuse. The hypotenuse is always opposite the right angle and always the longest side. It always goes on its own side of the equation (c² = a² + b²). Plugging a leg into the c position, or the hypotenuse into a leg position, gives a wrong answer every time.
Third, forgetting the square root at the end. When you calculate a² + b² = 25, the answer is not 25. That’s c squared. The actual side length is √25 = 5. This sounds obvious written out, but under time pressure on a test, it’s the most commonly skipped step.
Why This Formula Shows Up Everywhere
If you picture a square drawn on each side of a right triangle, the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides. That’s what a² + b² = c² actually means geometrically, and it’s the reason this relationship is so useful beyond math class.
In construction, the theorem is used to create precise right angles for building foundations. If you measure 3 feet along one edge and 4 feet along the other, the diagonal between those points should be exactly 5 feet if the corner is a true 90 degrees. Builders have used this technique, sometimes with knotted ropes, for thousands of years. Babylonian clay tablets from roughly 1900 to 1600 BCE already show knowledge of the theorem and lists of Pythagorean triples, more than a thousand years before Pythagoras was born.
Navigation relies on the same principle. A ship that has traveled 300 miles north and 400 miles west can calculate its straight-line distance from the starting point: 300² + 400² = 90,000 + 160,000 = 250,000, and √250,000 = 500 miles. GPS systems use this kind of distance calculation continuously. Pilots, engineers, machinists, and surveyors all depend on it. Any time you have a right angle and need to find a distance, the Pythagorean theorem is the tool.