A triangular prism and a pentagonal based pyramid would fit the given description.
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There is not such thing as a three-dimensional triangle. You are thinking of a tetrahedron which is a three-dimensional figure with each side being a triangle. This has four vertices.
A triangular prism has 5 faces and 6 vertices. At each vertex it has three plane angles - making 18 two-dimensional angles.
A polygon with three vertices and three sides is a triangle. A polygon with four vertices and four sides is a quadrilateral.
A pyramid is constructed from a square base (having four corners) and four triangles, rising from each side of the square with their tips meeting at one point. These five two-dimensional shapes form the faces of the three-dimensional shape of the pyramid, that therefore has five vertices (at the four corners of the square and at the point where the triangles meet).
You get the Volume by using formulas. There is usually a specific formula to find the volume of each shape. Some irregular shapes may not have a formula.