If the base of the pyramid is square, it must have four triangular vertical faces in order to meet at the apex (top). Such a 3D shape is a "square pyramid".
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A quadrilateral based pyramid, of which a square based pyramid would be a special case.
A square pyramid.
A square based pyramid has 5 corners - 4 for the square and 1 for the top of the pyramid.
A square based pyramid.