A pentagon in the plane with fixed side-lengths has a two-dimensional shape
space. Considering the pentagon as a mechanical system with point masses at the
corners we answer the question of how much the pentagon can rotate with zero
angular momentum. We show that the shape space of the equilateral pentagon has
genus 4 and find a fundamental region by discrete symmetry reduction with
respect to symmetry group D_5. The amount of rotation \Delta\theta, for a loop
in shape space at zero angular momentum is interpreted as a geometric phase and
is obtained as an integral of a function B over the region of shape space
enclosed by the loop. With a simple variational argument we determine locally
optimal loops as the zero contours of the function B. The resulting shape
change is represented as a Fourier series, and the global maximum of \Delta
\theta \approx 45\degree is found for a loop around the regular pentagram.