Elongated triangular pyramid

File:Tetraedro elongado 3D.stl

Johnson solid Template:Math.

In geometry, the elongated triangular pyramid is one of the Johnson solids (Template:Math). 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.Template:R 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.Template:R 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 $J_7$.Template:R

Properties

An elongated triangular pyramid with edge length $a$ has a height, by adding the height of a regular tetrahedron and a triangular prism:Template:R $$(1+\frac{\sqrt{6}}{3})a\approx 1.816a.$$ Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares:Template:R $$(3+\sqrt{3})a^2\approx 4.732a^2,$$ and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up:Template:R: $$\left(\frac{1}{12}(\sqrt{2}+3\sqrt{3})\right)a^3\approx 0.551a^3.$$

It has the three-dimensional symmetry group, the cyclic group $C_{3\mathrm{v}}$ of order 6. Its dihedral angle can be calculated by adding the angle of the tetrahedron and the triangular prism:Template:R

• the dihedral angle of a tetrahedron between two adjacent triangular faces is $\arccos \left(\frac{1}{3}\right)\approx 70.5^{\circ }$;
• the dihedral angle of the triangular prism between the square to its bases is $\frac{\pi }{2}=90^{\circ }$, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is $\arccos \left(\frac{1}{3}\right)+\frac{\pi }{2}\approx 160.5^{\circ }$;
• the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle $\frac{\pi }{3}=60^{\circ }$.

External links

• Template:Mathworld2