Biaugmented triangular prism

Biaugmented triangular prism
Type	Johnson J49 – J50 – J51
Faces	10 triangles 1 square
Edges	17
Vertices	8
Vertex configuration	${\displaystyle 2\times (3^{5})+2\times (3^{4})+4\times (3^{3}\times 4)}$
Symmetry group	${\displaystyle C_{2\mathrm {v} }}$
Properties	convex
Net

In geometry, the biaugmented triangular prism is a polyhedron constructed from a triangular prism by attaching two equilateral square pyramids onto two of its square faces. It is an example of Johnson solid.

Construction

The biaugmented triangular prism can be constructed from a triangular prism by attaching two equilateral square pyramids onto its two square faces, a process known as augmentation.^[1] These square pyramid covers the square face of the prism, so the resulting polyhedron has 10 equilateral triangles and 1 square as its faces.^[2] A convex polyhedron in which all faces are regular is Johnson solid, and the biaugmented triangular prism is among them, enumerated as 50th Johnson solid ${\displaystyle J_{50}}$.^[3]

Properties

A biaugmented triangular prism with edge length ${\displaystyle a}$ has a surface area, calculated by adding ten equilateral triangles and one square's area:^[2]

$${\displaystyle A={\frac {2+5{\sqrt {3}}}{2}}a^{2}\approx 5.3301a^{2}.}$$

Its volume can be obtained by slicing it into a regular triangular prism and two equilateral square pyramids, and adding their volumes subsequently:^[2]

$${\displaystyle V={\sqrt {{\frac {59}{144}}+{\frac {1}{\sqrt {6}}}}}a^{3}\approx 0.904a^{3}.}$$

It has three-dimensional symmetry group of the cyclic group ${\displaystyle C_{2\mathrm {v} }}$ 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, between triangle-to-triangle on the edge where an equilateral square pyramid and a triangular prism is attached, and between triangle-to-triangle on the edge where two square pyramids and a triangular prism are attached, is:^[4]

$${\displaystyle {\begin{aligned}\arccos \left(-{\frac {1}{3}}\right)+{\frac {\pi }{3}}&\approx 104.5^{\circ },\\\arccos \left(-{\frac {1}{3}}\right)+{\frac {\pi }{2}}&\approx 144.5^{\circ },\\2\arctan \left({\sqrt {2}}\right)+{\frac {\pi }{3}}&\approx 169.4^{\circ }.\end{aligned}}}$$

See also

• Augmented triangular prism — the 39th Johnson solid, constructed by attaching a triangular prism with one equilateral square pyramid.
• Triaugmented triangular prism — the 51st Johnson solid, constructed by attaching a triangular prism with three equilateral square pyramids.

References

External links

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