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"
A four faced figure is a tetrahedron. If they are all congruent triangles, they are all equilateral triangles and it is a regular tetrahedron - one of the 5 Platonic Solids.
A regular octahedron.
Octohedron, or rhomboid.
You're an octagon.
No, the bases are triangles, the sides are rectangles.
A true triangle is a two dimensional object, so it has NO faces at all (or maybe one). Triangles have 3 sides, but not 3 faces.