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Geometry · 02 · The famous one · 8 min

The Pythagorean theorem

In any right triangle, the square on the long side exactly equals the two smaller squares combined: a² + b² = c². It's the most used theorem in mathematics — and you can watch it be true.

Build the intuition

It's about actual squares

The equation isn't abstract: draw a real square on each side of a right triangle and the two smaller areas physically add up to the big one. The algebra a² + b² = c² is just bookkeeping for that picture.

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

From theorem to distance formula

How far is (1, 2) from (4, 6)? Walk 3 right and 4 up — two legs of a right triangle. The direct path is the hypotenuse: √(3² + 4²) = 5. Every “distance between points” in every dimension is Pythagoras wearing coordinates.

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Only right triangles need apply

The theorem is exclusive to 90° corners — and that exclusivity is a feature. Builders check that a corner is square by measuring 3, 4, then confirming the diagonal is 5. If it isn't, the corner isn't right. The theorem run backwards is a quality inspector.

See it move

InteractivePythagoras, watched live
abc
3
4
3² = 9 + 4² = 16 gives 25 — and the long side squared is 5² = 25. The two small squares always exactly fill the big one.

Resize the legs and watch the areas do their bookkeeping: the two small squares always exactly fund the big one.

A worked example

Will the ladder reach?

  1. A 5 m ladder leans with its foot 1.5 m from the wall. How high does it reach?

  2. Ladder = hypotenuse. Set up:

    1.52+h2=521.5^2 + h^2 = 5^2

  3. Solve:

    h=25−2.25=22.75≈4.77 mh = \sqrt{25 - 2.25} = \sqrt{22.75} \approx 4.77 \text{ m}

  4. Just under 4.8 m — checked before climbing, not after.

Out in the world

Your screen's diagonal

A “15-inch” laptop is measured corner to corner — the hypotenuse of the width-height triangle. Same with TVs. You've been buying hypotenuses your whole life.

Common confusion, cleared

“a² + b² = c² works for any triangle.”

Right triangles only. For others the diagonal is longer or shorter than √(a²+b²) — trigonometry's law of cosines handles those.

“c is just “the third side.””

c is specifically the hypotenuse — the side opposite the right angle, always the longest. Mixing it up with a leg is the classic exam slip.

Check yourself

PracticeQuick check

  1. Legs 6 and 8 — the hypotenuse is…

  2. How far is (0,0) from (5,12)?

Recap

  • In right triangles, the leg-squares exactly fill the hypotenuse-square.
  • The distance formula is Pythagoras in coordinates.
  • It runs backwards too: verify a right angle by checking the numbers.