Around 800 BCE, Baudhāyana worked out how to double a square, discovered the number we now call √2, and found the rule connecting the three sides of a right triangle — centuries before Pythagoras. Click through the tabs to rebuild his constructions, one slider at a time.
Doubling the side gives 4× the area — not double! Baudhāyana's trick: build a new square on the diagonal instead. Watch it split into triangles.
Reverse the idea: fold the corners of a square inward so the crease lines pass through the midpoints of each side. The tilted square PQRS has exactly half the area.
A unit square is 2 triangles; the square on its diagonal is 4 of the same triangles — double the area. That's how Baudhāyana found the length √2.
√2 has no exact decimal — but we can trap it between closer and closer bounds by squaring guesses. Slide to zoom in.
Euclid's classic argument (c. 300 BCE), the way Baudhāyana's discovery is completed:
Baudhāyana's general method: make a right triangle whose legs are the two square's sides. The square on its hypotenuse has area = sum of the two original areas.
The famous relation itself. Enter any two sides of a right triangle — legs or hypotenuse — and see the squares drawn to scale on each side.
Whole-number sidelengths that satisfy a² + b² = c² are called Baudhāyana triples. Enter three numbers to check, or list every triple up to a limit.
Since (n−1)² + (2n−1) = n², whenever the nth odd number (2n−1) is itself a perfect square, we get a brand-new Baudhāyana triple!
Inspired by Baudhāyana triples, Fermat asked: does xⁿ + yⁿ = zⁿ have whole-number solutions for n > 2? Try to find one.
A lotus stem pokes h units above a lake. A breeze pushes its tip d units sideways until it touches the water. How deep is the lake?
Not every square on a lattice has integer sides! Pick a "step" (across, up) between two dots — the tilted square you get has area = across² + up², even when the side itself is an irrational √N.
Test what you've learned across all twelve stations.