  This article focuses on the problem of analytically determining the optimal
placement of five points on the unit sphere $\mathbb{S}^2$ so that the surface
area of the convex hull of the points is maximized. It is shown that the
optimal polyhedron has a trigonal bipyramidal structure with two vertices
placed at the north and south poles and the other three vertices forming an
equilateral triangle inscribed in the equator. This result confirms a
conjecture of Akkiraju, who conducted a numerical search for the maximizer. As
an application to crystallography, the surface area discrepancy is considered
as a measure of distortion between an observed coordination polyhedron and an
ideal one. The main result yields a formula for the surface area discrepancy of
any coordination polyhedron with five vertices.
