In geometry, the elongated triangular pyramid is one of the Johnson solids (J7). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self-dual.
Construction
The elongated triangular pyramid is constructed from a triangular prism by attaching regular tetrahedron onto one of its bases, a process known as elongation.[1] The tetrahedron covers an equilateral triangle, replacing it with three other equilateral triangles, so that the resulting polyhedron has four equilateral triangles and three squares as its faces.[2] A convex polyhedron in which all of the faces are regular polygons is called the Johnson solid, and the elongated triangular pyramid is among them, enumerated as the seventh Johnson solid .[3]
Properties
An elongated triangular pyramid with edge length has a height, by adding the height of a regular tetrahedron and a triangular prism:[4]
Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares:[2]
and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up:[2]:
the dihedral angle of a tetrahedron between two adjacent triangular faces is ;
the dihedral angle of the triangular prism between the square to its bases is , and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is ;
the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle .