The Pythagorean Theorem
The most used formula in history: from ancient construction sites to modern GPS
Start discovering ↓Pythagoras rules the real world
Before talking about triangles and formulas: where do you encounter it every day?
Construction & Architecture
The 3-4-5 rule: ancient Egyptians used ropes knotted at 3-4-5 distances to construct perfect right angles in pyramids and temples.
GPS & Navigation
Your phone uses the Pythagorean theorem (and its 3D extension) to calculate your exact position within seconds.
3D Graphics & Video Games
Every distance calculated in a game engine, every collision detection, every shadow cast goes through a² + b² = c².
Navigation & Cartography
From Greek sailors to modern navigation systems: the distance between two points on a map is calculated with Pythagoras.
Medicine & Diagnostics
Ultrasound and CT scans use Pythagorean triangulation to reconstruct 3D images of the body interior.
Aviation & Routes
Pilots calculate flight routes and distances with 3D extensions of the theorem, accounting for altitude, latitude, and longitude.
A 4000-year-old story
1800 BC — Babylon
Cuneiform tablets already show Pythagorean triples like 3-4-5 and 5-12-13. Babylonian scribes used them to solve practical measurement problems.
2600 BC — Egypt
Craftsmen used ropes with 12 equally spaced knots, divided into 3-4-5 segments, to construct perfect right angles during pyramid construction.
570 BC — Pythagoras of Samos
The Greek philosopher proves the theorem rigorously for the first time, providing a general mathematical proof valid for all right triangles.
Today
From Euclid to Einstein, the Pythagorean theorem remains fundamental: it underlies Euclidean geometry, special relativity and nearly all modern physics.
Explore the right triangle
Move the sliders and watch how the squares on the sides change. Verify that the sum of the areas of the squares on the legs equals the area of the square on the hypotenuse.
3² + 4² = 9 + 16 = 25.00 → c = 5.000
The water proof
This is one of the most intuitive demonstrations of the theorem. The squares built on legs a and b contain the same amount of water as the square built on hypotenuse c.
Press "Pour the water" and watch water flow from squares a² and b² to fill exactly c². Not a drop left over, not a drop missing.
The 3-4-5 triangle is the oldest known example. The water in squares a²=9 and b²=16 fills exactly square c²=25.
Find the missing side
The theorem is used in three ways: find the hypotenuse given the legs, or find a leg given the hypotenuse and the other leg.
c = √(3² + 4²) = √25 ≈ 5.000
c ≈ 5.000
Classic proofs
Square proof
Build a square of side (a+b). Inside, place 4 identical right triangles. The remaining space is c². But a² + b² fills the same space too: therefore a² + b² = c².
Euclid's proof (Book I, Prop. 47)
Euclid proved the theorem in "Elements" using only axioms of plane geometry, without measurement. One of the most elegant and rigorous proofs in the history of mathematics.
Piece rearrangement
Cut the square on a into suitable pieces, then rearrange to exactly fill the square on c (minus square b²). This demonstrates that area is conserved.
The theorem, at last
Statement
In a right triangle, the square built on the hypotenuse equals the sum of the squares built on the two legs.
a² + b² = c²
where c is the hypotenuse (the side opposite the right angle) and a, b are the legs.
The converse is true
If in a triangle a² + b² = c² holds, then the triangle is right-angled. This is the "Pythagorean criterion" for verifying right angles.
a² + b² = c² ⟺ right angle
This allows verifying whether an angle is exactly 90° without measuring it directly.
3D Generalization
In three dimensions, the distance between two points is calculated by extending the theorem: apply Pythagoras twice, once in the plane, once in space.
d = √(a² + b² + c²)
Essential for GPS, 3D graphics, space navigation.
Test your knowledge!
A right triangle has legs 5 and 12. What is the hypotenuse?