Elongated square pyramid

In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid.

The elongated square pyramid is a composite, since it can be constructed by attaching one equilateral square pyramid onto one of the faces of a cube, a process known as elongation of the pyramid.^{[2] [3]} One square face of each parent body is thus hidden, leaving five squares and four equilateral triangles as faces of the composite.^[4]

A convex polyhedron in which all of its faces are regular is a Johnson solid, and the elongated square bipyramid is one of them, denoted as ${\displaystyle J_{8}}${\displaystyle J_{8}}, the fifteenth Johnson solid.^[5]

Given that ${\displaystyle a}${\displaystyle a} is the edge length of an elongated square pyramid. The height of an elongated square pyramid can be calculated by adding the height of an equilateral square pyramid and a cube. The height of an elongated square pyramid ${\displaystyle h}${\displaystyle h} (i.e., the distance between the square pyramid's apex and the center of a square base) is the sum of the cube's side and the height of an equilateral square pyramid. Its surface area ${\displaystyle A}${\displaystyle A} is the sum of four equilateral triangles and four squares' area. Its volume ${\displaystyle V}${\displaystyle V} is the sum of an equilateral square pyramid and a cube's volume. With edge length ${\displaystyle a}${\displaystyle a}, the formulation for each is:^{[6] [4]} $${\displaystyle {\begin{aligned}h&=\left(1+{\frac {\sqrt {2}}{2}}\right)a\approx 1.707a,\\A&=\left(5+{\sqrt {3}}\right)a^{2}\approx 6.732a^{2},\\V&=\left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\approx 1.236a^{3}.\end{aligned}}}$${\displaystyle {\begin{aligned}h&=\left(1+{\frac {\sqrt {2}}{2}}\right)a\approx 1.707a,\\A&=\left(5+{\sqrt {3}}\right)a^{2}\approx 6.732a^{2},\\V&=\left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\approx 1.236a^{3}.\end{aligned}}}

The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group ${\displaystyle C_{4v}}${\displaystyle C_{4v}} of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube:^[7]

• The dihedral angle of an elongated square pyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces, ${\displaystyle \arccos(-1/3)\approx 109.47^{\circ }}${\displaystyle \arccos(-1/3)\approx 109.47^{\circ }},
• The dihedral angle of an elongated square pyramid between two adjacent squares is the dihedral angle of a cube between those, ${\displaystyle \pi /2=90^{\circ }}${\displaystyle \pi /2=90^{\circ }},
• The dihedral angle of an equilateral square pyramid between a square and a triangle is ${\displaystyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }}${\displaystyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }}. Therefore, the dihedral angle of an elongated square pyramid between triangle-to-square, on the edge where the equilateral square pyramids attach the cube, is $${\displaystyle \arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.74^{\circ }.}$${\displaystyle \arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.74^{\circ }.}

• Elongated square bipyramid

• Weisstein, Eric W., "Johnson solid" ("Elongated square pyramid") at MathWorld .

• Composite polyhedron
• Johnson solids
• Self-dual polyhedra
• Pyramids (geometry)

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