A triangular prism can have a square or triangular base.
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Triangular
You can call it a "triangular prism" rather than "triangular based prism" ... saying a prism is "triangular" imply that the base is a triangle. There's no other name that I know of. A prism is just a 3d shape, so all you have to say is <type of shape> + prism to get the point across.
A triangular PYRAMID has a triangular base.
A triangular prism has triangular bases, a heagonal prism has ... you guessed it! ... hexagonal bases.
Every prism has a base. For a triangular prism the base is a triangle.