A pyramid.
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The solid shape that has one square face and four triangular faces is called a pyramid. If you were to have a shape that had four triangular faces and no square face, it would be called a triangular pyramid.
A square pyramid has 1 more face than a tetrahedron. A tetrahedron has 4 faces (equilateral triangles), while a square pyramid has 5 faces (4 triangles and 1 square base). Therefore, the square pyramid has one additional face compared to the tetrahedron.
A cube has 6 faces and a triangular pyramid has 5 faces. So a cube has 1 more face than a triangular prism. But a triangular pyramid has four faces. You make the call. That is correct: A triangular prism has five faces, and a cube has six faces, but a triangular pyramid has four faces.
A square pyramid has 5 faces. It has 1 square face and 4 triangle faces.
A square pyramid has 5 faces (4 triangular, 1 square base) and 5 vertices.