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"
square pyramid
an octagon
The answer is a pyramid.
octagonal pyramid :)
A pyramid.
The answer to your question, is an Octahedron.
An octagonal pyramid
A pyramid.
An octagonal pyramid.
The figure is a pyramid and the base is a square.
A regular octagonal pyramid.