Augmented triangular prism: Difference between revisions

Revision as of 07:42, 15 May 2024

Revision as of 07:44, 15 May 2024

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It has [[Point groups in three dimensions|three-dimensional symmetry group]] of the cyclic group <math> C_{2\mathrm{v}} </math> of order 4. Its [[dihedral angle]] can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism in the following:{{r|johnson}}

* The dihedral angle of an augmented triangular prism between two adjacent triangles is that of an equilateral square pyramid between two adjacent triangular faces, <math display="inline"> \arccos \left(-1/3 \right) \approx 109.5^\circ </math>, and is that of the interior angle of a triangular prism <math> \pi/3 = 60^\circ </math>.

* The dihedral angle of an augmented triangular prism between two adjacent squares is that of a triangular prism between two lateral faces, the interior angle of a triangular prism <math> \pi/3 = 60^\circ </math>.

* The dihedral angle of an augmented triangular prism between square and triangle is the dihedral angle of a triangular prism between the base and its lateral face, <math display="inline"> \pi/2 = 90^\circ </math>

* The dihedral angle of an equilateral square pyramid between a triangular face and its base is <math display="inline"> \arctan \left(\sqrt{2}\right) \approx 54.7^\circ </math>. Therefore, the dihedral angle of an augmented triangular prism between a square (the lateral face of the triangular prism) and triangle (the lateral face of the equilateral square pyramid) on the edge where the equilateral square pyramid is attached to the square face of the triangular prism, and between two adjacent triangles (the lateral face of both equilateral square pyramids) on the edge where two equilateral square pyramids are attached adjacently to the triangular prism, are <math display="block"> \begin{align}