Elongated square pyramid

The elongated square pyramid is a composite, since it can constructed by attaching one equilateral square pyramid onto one of the faces of a cube, a process known as elongation of the pyramid.{{r|timofeenko-2010|rajwade}} One square face of each parent body is thus hidden, leaving five squares and four equilateral triangles as faces of the composite.{{r|berman}}

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 $J\_\{15\}$, the fifteenth Johnson solid.{{r|uehara}}

Given that $a$ 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 a cube is the same as the edge length of a cube's side, and the height of an equilateral square pyramid is $(1/\backslash sqrt\{2\})a$. Therefore, the height of an elongated square bipyramid is:{{r|pye}}

$$a\; +\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}a\; =\; \backslash left(1\; +\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}\backslash right)a\; \backslash approx\; 1.707a.$$

Its surface area can be calculated by adding all the area of four equilateral triangles and four squares:{{r|berman}}

$$\backslash left(5\; +\; \backslash sqrt\{3\}\backslash right)a^2\; \backslash approx\; 6.732a^2.$$

Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them:{{r|berman}}

$$\backslash left(1\; +\; \backslash frac\{\backslash sqrt\{2\}\}\{6\}\backslash right)a^3\; \backslash approx\; 1.236a^3.$$

File:Pirámide cuadrada elongada.stl

The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group $C\_\{4v\}$ of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube:{{r|johnson}}

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

• Elongated square bipyramid

{{Reflist|refs=

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• {{mathworld2 | urlname2 = ElongatedSquarePyramid | title2 = Elongated square pyramid| urlname = JohnsonSolid | title = Johnson solid}}

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