The length of the polygonal spiral is found by noting that the ratio of inradius to circumradius of a regular polygon of n sides is r/R = (cot(π/n))/(csc(π/n)) = cos(π/n). The total length of the spiral for an n-gon with side length s is therefore L | = | 1/2 s sum_(k = 0)^∞ cos^k(π/n) | = | s/(2[1 - cos(π/n)]). Consider the solid region obtained by filling in subsequent triangles which the spiral encloses.

continuous line illusion | nested polygon | rep-tile | Theodorus spiral