A Hexahedron has no triangular faces.
A Tetrahedron has 4 triangular faces.
A octahedron has 8 triangular faces.
A Icosahedron has 20 triangular faces.
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It can, but only if it a right triangular prism one of whose faces exactly matches the face of the square.
No, it is not.
No because a triangular based pyramid is a tetrahedron.
A pentagonal pyramid is a hexahedron with a pentagonal face.
It is a hexahedron: more specifically, a triangular bipyramid.