Elongated triangular pyramid

File:Tetraedro elongado 3D.stl

3D model of an elongated triangular pyramid

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.Template:R/superscript 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/superscript 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/superscript

Properties

The height of an elongated triangular pyramid $h$ (i.e., the distance between a regular tetrahedron's apex and the center of a triangular base) is the height of a regular tetrahedron and a triangular prism. Its surface area $A$ can be calculated by adding the area of all eight equilateral triangles and three squares. Its volume $V$ can be calculated by slicing it into a regular tetrahedron and a prism, and then adding their volumes together. With edge length $a$, the formulation for each is:Template:R/superscriptTemplate:R/superscript $$\begin{array}{rl}h& =(1+\frac{\sqrt{6}}{3})a\approx 1.816a,\\ A& =(3+\sqrt{3})a^2\approx 4.732a^2,\\ V& =\left(\frac{1}{12}(\sqrt{2}+3\sqrt{3})\right)a^3\approx 0.551a^3.\end{array}$$

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/superscript

• 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