It is the most famous equation in history. Long before Einstein wrote E = mc², the ancient world was obsessed with a² + b² = c².
If you remember middle school geometry, you know it as the rule for right triangles. But did you know that Pythagoras, the man it’s named after, likely didn’t invent it? Or that construction workers still use this 2,500-year-old math hack every single day to ensure buildings don’t collapse?
Whether you are solving a homework problem, buying a new TV, or building a deck, the Pythagorean Theorem is the hidden logic that holds the physical world together.
The Formula: How to Find the Hypotenuse
The theorem is deceptively simple. It applies only to Right Triangles (triangles with one 90-degree corner).
- a = The height (Leg 1)
- b = The base (Leg 2)
- c = The hypotenuse (The long diagonal side)
The rule states that if you square the two shorter sides and add them together, the result equals the square of the longest side.
If you know any two sides, you can always find the third. But doing the square roots manually is a pain.
Ditch the Textbook
Don’t struggle with square roots. Enter your side lengths below to solve for the hypotenuse (or any leg) instantly.
Pp Calculate HypotenuseReal World Use #1: The “3-4-5” Construction Hack
If you have ever watched a house being built, you might have wondered: “How do builders ensure the corners are exactly 90 degrees?” They don’t have giant protractors.
Instead, they use the Pythagorean Theorem. They call it the 3-4-5 Rule.
If you measure 3 feet along one wall and mark it. Then measure 4 feet along the other wall and mark it. If the distance between those two marks is exactly 5 feet, the corner is perfectly square.
- 3² (9) + 4² (16) = 5² (25).
- It works perfectly every time.
If the distance is 4 feet and 11 inches, the builder knows the wall is crooked and needs to be fixed immediately. (Note: Make sure your measurements are accurate—check our Length Converter if you are mixing units!).
Real World Use #2: Will That TV Fit?
TVs are measured diagonally. When you buy a “55-inch TV,” that number is the measurement from the bottom-left corner to the top-right corner.
But that doesn’t tell you if it will fit inside your entertainment center. You need the Width and Height (a and b).
Using our Pythagorean Calculator, you can work backward. If you know the diagonal is 55 inches and the height is roughly 27 inches, you can solve for “b” to find the exact width.
The Dark History of Pythagoras
We teach Pythagoras as a brilliant mathematician, which he was. But he was also… unusual.
He led a cult-like group called the Pythagoreans in ancient Greece. They worshipped numbers as gods. They believed that numbers were the building blocks of the soul.
Did he really invent it?
Probably not. Archeologists have found clay tablets from Ancient Babylon (modern-day Iraq) dating back to 1800 B.C.—that is 1,000 years before Pythagoras was born. These tablets clearly list “Pythagorean Triples,” proving that Babylonian engineers were using this math to build temples long before the Greeks claimed it.
Common Pythagorean Triples
A “Triple” is a set of three whole numbers that fit the equation perfectly (no decimals). Memorizing these will help you ace your next math test.
| Leg A | Leg B | Hypotenuse C |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
Conclusion: A Tool for Everything
From figuring out the safe angle to lean a ladder against a wall, to designing 3D graphics in video games, this ancient formula is everywhere.
You don’t need to be a cult leader to appreciate the elegance of a perfect right triangle. Use our calculator to solve your geometry problems in seconds.