Sounds like an octahedron.
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A regular octahedron - two square based pyramids stuck together along their square bases.
Pyramids are known with triangular and square bases. In theory, you could build a pyramid using any regular polygon as a base.
Main difference is that square pyramids have square bases, cones have circular bases. For this reason, they're calculated differently.
Properties of the Pyramids include square bases, triangular bases, equilateral triangles, isosceles triangles, interior right angle triangles and much much more.
Cubes are prisms that have squares as bases and the height is the same as the edges. A square prism has square bases, top and bottom, and parallelograms as sides. A square pyramid has a square base and triangles for the vertical sides.