What shape can have a triangular base and triangular faces?

• 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 triangles

• An octohedron (eight sides)
• An isocahedron (20 identical equilateral triangular faces, 30 edges and 12 vertices)
• Any of a number of deformations of the above