Elongated triangular pyramid
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In geometry, the elongated triangular pyramid is one of the Johnson solids (Template:Math). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self-dual.
ConstructionEdit
The elongated triangular pyramid is constructed from a triangular prism by attaching regular tetrahedron onto one of its bases, a process known as elongation.Template:R The tetrahedron covers an equilateral triangle, replacing it with three other equilateral triangles, so that the resulting polyhedron has four equilateral triangles and three squares as its faces.Template:R A convex polyhedron in which all of the faces are regular polygons is called the Johnson solid, and the elongated triangular pyramid is among them, enumerated as the seventh Johnson solid <math> J_7 </math>.Template:R
PropertiesEdit
An elongated triangular pyramid with edge length <math> a </math> has a height, by adding the height of a regular tetrahedron and a triangular prism:Template:R <math display="block"> \left( 1 + \frac{\sqrt{6}}{3}\right)a \approx 1.816a. </math> Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares:Template:R <math display="block"> \left(3+\sqrt{3}\right)a^2 \approx 4.732a^2, </math> and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up:Template:R: <math display="block"> \left(\frac{1}{12}\left(\sqrt{2}+3\sqrt{3}\right)\right)a^3 \approx 0.551a^3. </math>
It has the three-dimensional symmetry group, the cyclic group <math> C_{3\mathrm{v}} </math> of order 6. Its dihedral angle can be calculated by adding the angle of the tetrahedron and the triangular prism:Template:R
- the dihedral angle of a tetrahedron between two adjacent triangular faces is <math display="inline"> \arccos \left(\frac{1}{3}\right) \approx 70.5^\circ </math>;
- the dihedral angle of the triangular prism between the square to its bases is <math display="inline"> \frac{\pi}{2} = 90^\circ </math>, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is <math display="inline"> \arccos \left(\frac{1}{3}\right) + \frac{\pi}{2} \approx 160.5^\circ </math>;
- the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle <math display="inline"> \frac{\pi}{3} = 60^\circ </math>.
ReferencesEdit