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Lesson 11 of 20 · 3 moves

The Root of Two

Swing the square’s diagonal down onto the line of its base.

ABCD
The finished construction: the figure in gold over the lines that found it.

The diagonal of a unit square is √2. Legend says the Pythagorean who proved it irrational was drowned for revealing the secret.

Draw the diagonal from A, then carry its length down with a circle so it lies along the base.

Try it in Construct

The app snaps to every crossing, checks your construction exactly, and offers hints one move at a time.

Step by step

  1. ABCD

    Draw the diagonal from A to C.

  2. ABCD

    Extend the base AB into a line.

  3. ABCD

    Circle centred on A through C: it lands on the base line at √2.

What it shows

AE is √2 times AB. Fold a sheet of A4 paper in half and it keeps this same ratio: that is why the paper sizes use it.