A pentagonal prism has 10 vertices because there are two five-sided shapes at both ends of a prism. A pentagonal prism has 15 edges and seven faces. Five faces are between the pentagons on each end.
Comparatively, a rectangular prism has eight vertices, 12 edges and six faces. A triangular prism contains six vertices, nine edges and five faces. A hexagonal prism has 12 vertices, 18 edges and eight faces. Every time an edge of a regular polygon is added to a three-dimensional prism, the vertices increase by two, the edges increase by three and the faces increase by one. The reverse is true when an edge is subtracted from a regular polygon in a prism-like figure.