Spiral of Theodorus — Definition, Formula & Examples

The Spiral of Theodorus is a spiral formed by placing right triangles end-to-end, where each triangle has a short leg of length 1 and a hypotenuse that becomes the long leg of the next triangle. The hypotenuses have lengths

\sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \ldots

, creating a visual representation of successive square roots.

The Spiral of Theodorus is a sequence of contiguous right triangles $T_1, T_2, T_3, \ldots$ sharing a common vertex at the origin, where triangle $T_n$ has legs of length $\sqrt{n}$ and $1$ , yielding a hypotenuse of length $\sqrt{n+1}$ . The hypotenuse of $T_n$ serves as the longer leg of $T_{n+1}$ , and the outer vertices trace a spiral curve.

Key Formula

h_n = \sqrt{n+1}

Where:

$h_n$ = Length of the hypotenuse of the nth triangle
$n$ = Triangle number (starting from 1)

How It Works

Start by drawing a right triangle with both legs equal to 1, giving a hypotenuse of

\sqrt{2}

. Next, build a new right triangle on that hypotenuse: use

\sqrt{2}

as one leg and 1 as the other, producing a hypotenuse of

\sqrt{3}

. Continue this process indefinitely. Each new hypotenuse equals

\sqrt{n+1}

, where

n

is the triangle number. The triangles share a common vertex at the center and fan outward, creating a spiral pattern. After roughly 17 triangles, the spiral completes its first full turn.

Worked Example

Problem: Find the hypotenuse lengths of the first four triangles in the Spiral of Theodorus.

Triangle 1: Both legs are 1. Apply the Pythagorean theorem.

h_1 = \sqrt{1^2 + 1^2} = \sqrt{2}

Triangle 2: The long leg is

\sqrt{2}

(previous hypotenuse) and the short leg is 1.

h_2 = \sqrt{(\sqrt{2})^2 + 1^2} = \sqrt{3}

Triangle 3: The long leg is

\sqrt{3}

and the short leg is 1.

h_3 = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{4} = 2

Triangle 4: The long leg is 2 and the short leg is 1.

h_4 = \sqrt{2^2 + 1^2} = \sqrt{5}

Answer: The first four hypotenuses are

\sqrt{2}, \sqrt{3}, 2, \sqrt{5}

Why It Matters

The Spiral of Theodorus gives a concrete geometric proof that

\sqrt{n}

exists for every positive integer, which was a profound insight for ancient Greek mathematicians. It connects the Pythagorean theorem to irrational numbers and spiral geometry, topics that appear in high school geometry and precalculus courses.

Common Mistakes

Mistake: Using the previous hypotenuse as the new hypotenuse instead of as the new leg.

Correction: Each hypotenuse becomes the longer leg of the next triangle. The new hypotenuse is always computed from the Pythagorean theorem using that leg and a short leg of 1.