Shape and space

c. 1800 BC

Right triangles in clay

Babylonian scribes

How do you make a perfectly square corner with nothing but a rope?

A clay tablet from Babylon, now called Plimpton 322, lists rows of numbers that fit right-angled triangles. It was written over a thousand years before Pythagoras was born. Nobody is sure what it was for: a teacher’s table, or a tool for builders.

How they figured it out

Babylonian scribes counted in base 60 and loved tables: reciprocals, squares, interest. Plimpton 322 lists fifteen rows of number pairs whose squares differ by a perfect square, neatly sorted from a steep slope to a shallow one, including the triple 119, 120 and 169.

Historians still argue about what it was for. A teacher’s answer key? A builder’s reference? What nobody disputes is that someone had a recipe for generating these triples and worked out fifteen of them, some in the thousands, without an error we can find.

The equation

a2+b2=c2a^2 + b^2 = c^2

In a right-angled triangle, the square on the longest side equals the two other squares added together.

Everyday echoBuilders still check a corner is square with a 3–4–5 triangle.

See it in real life

The rope trick

Tie a loop of rope with 12 equally spaced knots and pull it into a triangle with sides of 3, 4 and 5 knots. The corner between the 3 and the 4 is exactly square. Builders still check a floor this way: 3 m along one wall, 4 m along the other, and the diagonal must be 5 m.

Where engineers use it

  • Setting out buildings

    Surveyors check corners with a 3–4–5 triangle; laser levels and total stations still rely on the same rule.

  • Distances on screens

    Every game engine and map app finds straight-line distance with Δx2+Δy2\sqrt{\Delta x^2 + \Delta y^2}.

Worked example: A ladder against a wall

  1. A 5 m ladder with its foot 3 m from the wall.
  2. Height reached: 52−32=16=4\sqrt{5^2 - 3^2} = \sqrt{16} = 4 m.

What it led to