General hexahedral and quadrangular grids present a challenge for mixed finite elements for second-order, elliptic problems. We define and analyze a mixed finite element method for a mesh made up of star-shaped polygons. The scalar unknown is approximated by element-wise constants and the vector unknown is determined by its flux through the edges of the polygons. The elements are composite elements. Each polygon is split into triangles by taking an interior point of the polygon, one for which it is star-shaped, and considering the triangles radiating from that point and having one side as a side of the polygon. Convergence of the method is proven, and numerical experiments are shown to confirm the theoretical results.