MathDB

The cross of a convex

n

-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a

3 \times 4

rectangle is

\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}

.
Suppose

S

is a dodecagon (

12

-gon) inscribed in a unit circle. Find the greatest possible cross of

S

Problem Statement