Pythagorean theorem formula
The Pythagorean theorem relates the three sides of a right triangle. The square of the hypotenuse — the side opposite the right angle — equals the sum of the squares of the other two sides. It is the foundation of distance and geometry calculations.
What each symbol means
| Symbol | Meaning | Units |
|---|---|---|
| a | One leg of the right angle | any length unit |
| b | The other leg | same unit |
| c | Hypotenuse (longest side) | same unit |
Rearranged forms
Worked example
A right triangle has legs of 3 and 4. Find the hypotenuse.
- Start from c² = a² + b².
- Substitute a = 3 and b = 4: c² = 9 + 16.
- Add: c² = 25.
- Take the square root: c = √25.
All three sides must use the same length unit. The theorem only applies to right triangles; for other triangles use the law of cosines. The 3-4-5 and 5-12-13 are handy integer right triangles to remember.
Have two sides and need the third?
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How the theorem works
Build a square on each side of a right triangle. The theorem says the area of the square on the hypotenuse exactly equals the combined area of the squares on the two legs. That geometric fact becomes the algebra c² = a² + b².
Where it is used
The theorem underpins the distance formula, navigation, construction layout (the 3-4-5 method for square corners), screen and TV diagonals, and any time you need a straight-line distance from horizontal and vertical amounts.
FAQ
What is the Pythagorean theorem?
For a right triangle, c² = a² + b², where c is the hypotenuse and a and b are the two legs.
Does it work for any triangle?
No, only right triangles. For triangles without a right angle, use the law of cosines, which adds a correction term.
What is a Pythagorean triple?
A set of three whole numbers that satisfy the theorem, such as 3-4-5 or 5-12-13. They form right triangles with integer sides.