Elongated square pyramid
In geometry, the elongated square pyramid is a convex composite polyhedron constructed by attaching the square base of an equilateral square pyramid to one of the square faces of a cube; this is called an elongation of the pyramid. One square face of each parent body is thus hidden, leaving five squares and four equilateral triangles as faces of the composite.
It is an example of a Johnson solid, a convex polyhedron whose faces are all regular, indexed as.
Properties
The height of an elongated square pyramid is the sum of the cube's side and the height of an equilateral square pyramid. Its surface area is the sum of four equilateral triangles and four squares' area. Its volume is the sum of an equilateral square pyramid and a cube's volume. With edge length, the formulation for each is:The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group of order eight.
It has three kinds of dihedral angle:
- The dihedral angle between adjacent triangles is that of a regular octahedron,.
- The dihedral angle between adjacent squares is that of a cube,.
- In a square pyramid, the dihedral angle between the square base and a triangle side is ; thus in an elongated square pyramid the angle between a triangle and a square, on the edge where the pyramid attaches to the cube, is