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A pyramid is a structure with a square at the bottom and 4 triangles connecting them. You have the correct word.
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.
A triangular pyramid has 4 triangular sides.A rectangular pyramid has 4 triangular sides and 1 rectangular side.
a pyramid with a triangular base has 4 faces. a pyramid with a square base has 5 faces.
A tetrahedron is a kind of pyramid with 4 vertices. A tetrahedron pyramid is probably an object in 4-dimensional hyperspace.