Constructible Numbers

Lengths built from square roots

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Compass and straightedge can add, subtract, multiply, divide and take square roots.
• A semicircle on $a + b$ gives height $\sqrt{ab}$.
• Nothing else is possible: cube roots like ∛2 are out of reach.

Steps

1. Arithmetic with a compass
2. Build a root
3. A root from a triangle
4. The big theorem
5. Building √5
6. The spiral of Theodorus
7. Which length can you build?