tetrahedron aka triangular pyramid
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A pyramid has the same number of faces as vertices. A triangular pyramid has 4. A rectangular pyramid has 5. Check into Euler's Theorems.
cube
One possible pair is a cube and a parallelepiped. Vertices and corners are the same thing!
It is a cuboid or a cube
No faces, 1 edge, no corners. A circular disk has 1 face, but that is not the same as a circle.