square base pyramid and a triangular base pyramid
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Tetrahedron
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"
No.
A solid with four faces for maths is a tetrahedron.
A pyramid. The square being the base and the triangles the 'sides' - each having a side connected to a side of the square - the other sides connected to 'neighbor-triangles'.