  An equilateral pentagon is a polygon in the plane with five sides of equal
length. In this paper we classify the central configurations of the $5$-body
problem having the five bodies at the vertices of an equilateral pentagon with
an axis of symmetry. We prove that there are two unique classes of such
equilateral pentagons providing central configurations, one concave equilateral
pentagon and one convex equilateral pentagon, the regular one. A key point of
our proof is the use of rational parameterizations to transform the
corresponding equations, which involve square roots, into polynomial equations.
