square based pyramid, And vertexes is not a word, they are called a vertex(singular) or vertices(plural)
The tetrahedron, consisting of four regular triangles.
5 vertexes, 4 faces, and 8 edges
A square pyramid.
Pyramid
It is a triangular prism that has 5 faces. 6 vertices and 9 edges
square
An icosahedron has 20 faces, 30 edges, and 12 vertexes. 5 polygons meet at each vertex and each face has 3 vertexes (therefore made of triangles). A dodecahedron has 12 faces, 30 edges, and 20 vertexes. 3 polygons meet at each vertex and each face has 5 vertexes (therefore made of pentagons).
Two tetrahedron stuck together by a face.
Triangular prism of course!
square based Pyramid
A square based pyramid fits the given description
A square based pyramid has 5 vertices, 5 faces and 8 edges