A shape with a square base and four triangular sides is a pyramid.
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There is no specific name because there is no specific shape. The shape could have anything from 5 to 13 sides depending on the sizes and orientations of the triangle and square.
It is either a square or an equilateral triangle.
A square pyramid's base is a square. You can tell by the name (for exp. a triangular pyramid's base is a triangle.) Good luck with whatever you need it for :)
A square based pyramid perhaps.
The simplest such shape is PYRAMID.