Spiral of Theodorus
Right triangles with legs √n and 1 stacked around a point: the hypotenuses are √2, √3, √4 … and the outer edge spirals out.
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Formulas and key values
- Construction rule
Put a leg of length 1 at right angles to the last hypotenuse, and join its end to the centre.
- Hypotenuse k
- Angle of triangle k
- Outer edge lengthIn the graph = 16
n is the number of right triangles; triangle k has legs √k and 1.
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Start the video01 · Definition
What is the Spiral of Theodorus?
Start with a right triangle whose legs are both 1; its hypotenuse is √2. Use that hypotenuse as one leg of the next right triangle, with the other leg 1 again, and its hypotenuse is √3. Keep going and the hypotenuses run through √2, √3, √4, √5 … while the outer edges coil into a spiral.
It is named after Theodorus of Cyrene, who, according to Plato’s Theaetetus, proved that the square roots of 3, 5, … up to 17 are irrational. One explanation for stopping at √17: the triangle with that hypotenuse is the last one before the spiral overlaps itself.
02 · Properties
Key properties
- The k-th hypotenuse has length √(k + 1).
- Triangle k adds the angle arctan(1/√k) at the centre.
- The outer edge after n triangles has length n.
- The first 16 triangles (up to hypotenuse √17) turn 351°; the 17th overlaps the first.
- Successive turns approach a constant spacing, close to π.
03 · Questions
Spiral of Theodorus: common questions
- How is the Spiral of Theodorus constructed?
- Construction rule: Put a leg of length 1 at right angles to the last hypotenuse, and join its end to the centre.. n is the number of right triangles; triangle k has legs √k and 1.
04 · Keep exploring