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A pyramid can have a polygon with any number of sides as its base. A triangular pyramid (tetrahedron, the Platonic solid) and the rectangular based pyramid (like the Egyptian ones) are better known, but that is simply a result of poor imagination.
If a pyramid has an n-gon as its base then it will have:
2 * n edges
n + 1 faces
and n + 1 vertices.
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