A pyramid with a triangular base. The faces are 4 equilateral triangles.
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A pyramid
A square is a rectangle with 4 equal sides and 4 equal angles. A rhombus is a parallelogram with 4 equal sides.
Rectangle.
Square
There is a contradiction in your question. If the base has four equal sides then is it a square so not all the faces are Triangles. However if the base is a square and the rest of the faces are triangles then your object is a "Square-based Pyramid"