The solid commonly called a dodecahedron is not a solid pyramid but the name simply means a solid shape with 12 faces and so, a pyramid on a 11-sided polygonal base will be a dodecahedron.
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A pyramid has the solid quadrilateral pyramid solid shape.However there are also pyramids with geometrical polyhedra/platonic solid shapes of tetrahedron.
Square Pyramid
The number of vertices on a solid pyramid will depend on its base as for example if it is a triangular based pyramid (tetrahedron) then it will have 4 vertices, 6 edges and 4 faces.
a square pyramid
Yes.