A tetrahedron is a kind of pyramid with 4 vertices. A tetrahedron pyramid is probably an object in 4-dimensional hyperspace.
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Triangular pyramid = tetrahedron has 4 points = 4 vertices
A tetrahedron which is a triangular based pyramid has 4 vertices, 4 faces and 6 edges
A triangular based pyramid which is a tetrahedron has 4 faces, 6 edges and 4 vertices
A tetrahedron is a triangular based pyramid that has 6 edges, 4 faces and 4 vertices
A tetrahedron is a triangular based pyramid that has 4 faces, 6 edges and 4 vertices