Pythagorean Theorem Guide: Proof, Formula & 3D
Pythagorean theorem a² + b² = c² with proofs, 3D extension, real-world examples, and calculator. Complete math guide with worked problems and practice sets.
Pythagorean Theorem — Direct Answer
The Pythagorean theorem states that in any right triangle, a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle) and a, b are the legs. If legs are 3 and 4, the hypotenuse is 5. It works in any unit and generalizes to 3D as d² = a² + b² + c² for space diagonals.
→ Solve any right triangle with our free triangle calculator
How to Use the Pythagorean Theorem
Three versions of the same formula cover every right triangle problem.
Find hypotenuse: c = √(a² + b²)
Find a leg: a = √(c² − b²)
Find b leg: b = √(c² − a²)
Where:
- a, b = legs (the two shorter sides)
- c = hypotenuse (longest side, opposite right angle)
How to identify a right triangle
A right triangle has one 90° angle. The side opposite this angle is the hypotenuse (always the longest). The two sides forming the 90° angle are the legs.
How to apply the formula to find the hypotenuse
Example: legs 6 and 8
- c² = 6² + 8²
- c² = 36 + 64 = 100
- c = 10
How to apply the formula to find a leg
Example: hypotenuse 13, one leg 5
- b² = c² − a²
- b² = 169 − 25 = 144
- b = 12
How to check if a triangle is right
Use the converse: if a² + b² = c², the triangle is right. If not, it's acute or obtuse.
Example: sides 7, 24, 25
- 49 + 576 = 625 = 25² ✓ Right triangle
Example: sides 5, 6, 7
- 25 + 36 = 61 ≠ 49 ✗ Not right
Pythagorean Theorem Chart & Triples
Common Pythagorean triples
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 6 | 8 | 10 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
| 9 | 12 | 15 |
| 9 | 40 | 41 |
| 11 | 60 | 61 |
| 12 | 16 | 20 |
| 12 | 35 | 37 |
| 20 | 21 | 29 |
Right triangle side reference
| Leg a | Leg b | Hypotenuse c |
|---|---|---|
| 1 | 1 | 1.414 |
| 1 | 2 | 2.236 |
| 2 | 3 | 3.606 |
| 3 | 4 | 5.000 |
| 5 | 5 | 7.071 |
| 6 | 8 | 10.000 |
| 10 | 10 | 14.142 |
Proof of the Pythagorean Theorem
Algebraic proof (rearrangement)
Take 4 identical right triangles with legs a, b, and hypotenuse c. Arrange them in a square of side (a + b). The inner square has side c and area c². The 4 triangles total 2ab. The big square has area (a + b)² = a² + 2ab + b². Setting equal: a² + 2ab + b² = c² + 2ab → a² + b² = c² ∎
Geometric proof (similar triangles)
Drop altitude to hypotenuse. This creates 3 similar triangles. Side ratios produce a² + b² = c² after algebra.
Historical origin
Named after Pythagoras (~570 BC), though evidence suggests Babylonians knew it 1,000 years earlier (Plimpton 322 tablet).
Pythagorean Theorem in 3D
The theorem extends to three dimensions for space diagonals.
Space diagonal: d = √(a² + b² + c²)
Where:
- a, b, c = three edges of a rectangular box
- d = diagonal from corner to opposite corner
3D example: box diagonal
Box 3 × 4 × 12:
- d = √(9 + 16 + 144) = √169 = 13
3D in video games and physics
- Distance between objects in 3D space
- Vector magnitude
- Force calculations
- Collision detection
Pythagorean Theorem Applications
Construction
- Squaring foundations (3-4-5 rule)
- Rafter length calculation
- Roof slope calculation
- Diagonal bracing
Navigation
- GPS distance between waypoints
- Map distances
- Flight paths
Video games
- Distance between players
- Collision detection
- Sprite rendering
Physics
- Velocity vector magnitude
- Resultant forces
- Electric field strength
Common Pythagorean Theorem Mistakes
Mistake 1: Using it on non-right triangles
The theorem only works for right triangles (90° angle). Fix: For non-right triangles, use the law of cosines: c² = a² + b² − 2ab·cos(C).
Mistake 2: Forgetting to square root
Many people stop at a² + b² = 25 and give the answer as 25, not 5. Fix: Final step is √. 25 → 5.
Mistake 3: Putting the hypotenuse on the wrong side
If c is a leg, the equation is wrong. c must be the hypotenuse. Fix: Identify the right angle first. The side opposite is the hypotenuse.
Pythagorean Theorem FAQ
What is the Pythagorean theorem?
a² + b² = c². In any right triangle, the sum of the squares of the legs equals the square of the hypotenuse.
How do you find the hypotenuse?
c = √(a² + b²). Square both legs, add them, then take the square root.
How do you find a missing leg?
a = √(c² − b²). Rearrange the formula to isolate the missing side.
What are Pythagorean triples?
Whole-number right triangles like 3-4-5, 5-12-13, 8-15-17. There are infinitely many.
Does the Pythagorean theorem work in 3D?
Yes. d² = a² + b² + c² for the space diagonal of a 3D box.
Next Steps for Right Triangle Math
- Triangle Calculator — Any triangle
- Distance Formula Calculator — 2D coordinates
- Math Calculator — All math tools
- Distance Formula Guide — Coordinate math
- 3-4-5 Squaring Guide — Construction
Last updated October 2026. Works in any unit — meters, feet, inches, parsecs.