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an Egyptian pyramid
If by "side" you mean face, you might be referring to a heptagonal pyramid (a pyramid with eight faces).If by "side" you mean edge, you might be referring to a tetragonal pyramid (a pyramid with eight edges).Or, if you mean it has an eight-sided base, your answer would be an octagonal pyramid (a pyramid with an eight-sides base).
A headstone can be a rectangular pyramid.
If you mean the base has 4 sides then the pyramid is a square based pyramid and has 5 corners. If you mean the pyramid has 4 faces then the pyramid is a tetrahedron with a triangular base and has 4 corners.
The Egyptian pyramids.