A triangular based pyramid would seem to fit the given description which has 4 equal sided triangular faces, 6 edges and 4 vertices.
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"
The answer is a pyramid.
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The figure is a pyramid and the base is a square.
A tetrahedron.
octagonal pyramid
A pyramid with a triangular base. The faces are 4 equilateral triangles.
A regular pentagon-based pyramid.
Pyramid.
An octahedron
square pyramid
a pyramid, in the sense that most people think of has a square base, with 4 triangle 'sides', which would be 5 faces. A regular tetrahedron has 4 equilateral triangles as faces.