Biaugmented triangular prism

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]

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]

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

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

$${\displaystyle {\frac {59+24{\sqrt {6}}}{144}}a^{3}\approx 0.904a^{3}.}$$

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]

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

• 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.

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