Augmented triangular prism

Augmented triangular prism
Type	Johnson J48 – J49 – J50
Faces	3x2 triangles 2 squares
Edges	13
Vertices	7
Vertex configuration	2(3.42) 1(34) 4(33.4)
Symmetry group	C2v
Dual polyhedron	monolaterotruncated triangular bipyramid
Properties	convex
Net

In geometry, the augmented triangular prism is one of the Johnson solids (J₄₉). As the name suggests, it can be constructed by augmenting a triangular prism by attaching a square pyramid (J₁) to one of its equatorial faces. The resulting solid bears a superficial resemblance to the gyrobifastigium (J₂₆), the difference being that the latter is constructed by attaching a second triangular prism, rather than a square pyramid.

Construction

The augmented triangular prism can be constructed from a triangular prism by attaching an equilateral square pyramid to one of its square faces. This square pyramid covers the square face of the prism, so the resulting polyhedron has 6 equilateral triangles and 2 squares as its faces. A convex polyhedron in which all faces are regular is Johnson solid, and the augmented triangular prism is among them, enumerated as 49th Johnson solid $J_{49}$.

Properties

An augmented triangular prism with edge length $a$ has a surface area, calculated by adding six equilateral triangles and two squares' area: $${\frac {4+3{\sqrt {3}}}{2}}a^{2}\approx 4.598a^{2}.$$ Its volume can be obtained by slicing it into a regular triangular prism and an equilateral square pyramid, and adding their volume subsequently: $${\frac {2{\sqrt {2}}+3{\sqrt {3}}}{12}}a^{3}\approx 0.669a^{3}.$$

It has three-dimensional symmetry group of the cyclic group $C_{2\mathrm {v} }$ of order 4. Its dihedral angle can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism. The dihedral angle of an equilateral square pyramid between two adjacent triangular faces is ${\textstyle \arccos \left(-1/3\right)\approx 109.5^{\circ }}$, and that between a triangular face and its base is ${\textstyle \arctan \left({\sqrt {2}}\right)\approx 54.7^{\circ }}$. The dihedral angle of a triangular prism between two adjacent square faces is the internal angle of an equilateral triangle ${\textstyle \pi /3=60^{\circ }}$, and that between square-to-triangle is ${\textstyle \pi /2=90^{\circ }}$. Therefore, the dihedral angle of the augmented triangular prism between square-to-triangle and triangle-to-triangle on the edge where both square pyramid and triangular prism are attached is, respectively: $${\begin{aligned}{\frac {\pi }{3}}+\arccos \left(-{\frac {1}{3}}\right)&\approx 104.5^{\circ },\\{\frac {\pi }{2}}+\arccos \left(-{\frac {1}{3}}\right)&\approx 144.5^{\circ }.\end{aligned}}$$

External links

• Weisstein, Eric W. "Johnson Solid". MathWorld.
  • Weisstein, Eric W. "Augmented triangular prism". MathWorld.