Constructing 3D Figures from Polygons

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By 3d figure I mean what a pyramid with a triangular base is to a triangle and what a cube is to a square. My question is, is it possible to form a tridemensional figure with any polygon? If so, what is the relation between the number of sides of the polygon and the number of faces of the figure?
 
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This can only be done with the triangle, square, and pentagon. They are called the platonic solids
http://www.hitxp.com/math/geo/euclid/210503.htm
 
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