Augmented triangular prism

The augmented triangular prism can be constructed from a triangular prism by attaching an equilateral square pyramid to one of its square faces, a process known as augmentation.[1] This square pyramid covers the square face of the prism, so the resulting polyhedron has 6 equilateral triangles and 2 squares as its faces.[2] 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 ${\displaystyle J_{49}}$.[3]

An augmented triangular prism with edge length ${\displaystyle a}$ has a surface area, calculated by adding six equilateral triangles and two squares' area:[2]

$${\displaystyle {\frac {4+3{\sqrt {3}}}{2}}a^{2}\approx 4.598a^{2}.}$$

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

$${\displaystyle {\frac {2{\sqrt {2}}+3{\sqrt {3}}}{12}}a^{3}\approx 0.669a^{3}.}$$

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 and triangle-to-triangle on the edge where both square pyramid and triangular prism are attached is, respectively:[4]

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

In the geometry of chemical compounds, a polyhedron may commonly visualize an atom cluster surrounding a central atom. The capped trigonal prismatic molecular geometry describes clusters for which this polyhedron is an augmented triangular prism.[5] An example of such compound is the potassium heptafluorotantalate.[6]

• Biaugmented triangular prism — the 50th Johnson solid, constructed by attaching a triangular prism with two equilateral square pyramids.
• 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. "Augmented triangular prism". MathWorld.