Why No Hexagonal Pyramid?
Six equilateral triangles around one point lie perfectly flat and form a hexagon, so a pyramid on a hexagon cannot have equilateral sides: its apex would sit on the base. With a square or a pentagon at the bottom, the triangles still have room to rise to a point.
Where Is the Square Bipyramid?
Two square pyramids with equilateral sides, joined base to base, make the regular octahedron, one of the Platonic solids. Its net is on the page of Platonic solid nets, so this page keeps the triangular and pentagonal bipyramids.
Pyramids in the Classroom
A folded square pyramid gives children a model to hold next to pictures of the pyramids of Egypt, and to talk about a base, faces and an apex. Counting the triangles of a bipyramid also shows that a closed shape does not need a flat base to stand on.