That object would be called a triangular pyramid, also known as a tetrahedron.
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The 3D shape you are describing is a pentagonal pyramid. It has a pentagonal base with five sides, and five vertices where the triangular faces meet. The three rectangle faces are lateral faces that connect the triangular base to the apex of the pyramid. The pentagonal pyramid is a type of pyramid with a polygonal base and triangular faces.
A three dimensional triangular figure with three faces and a base is called a triangular pyramid. It has four vertices, four faces, and six edges.
The given description fits that of a triangular based pyramid which has 4 faces, 6 edges and 4 vertices
A triangular prism, a triangle base and three triangular faces (a tetrahedron)The figures below do not strictly have a "base" but they are composed entirely of trianglesAn octohedron (eight sides)An isocahedron (20 identical equilateral triangular faces, 30 edges and 12 vertices)Any of a number of deformations of the above
A rectangular pyramid has 5 faces: a rectangular base, and 4 triangular faces.