Editing Elongated square pyramid (section)

== Properties ==
Given that <math> a </math> 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 <math> (1/\sqrt{2})a </math>. Therefore, the height of an elongated square bipyramid is:{{r|pye}}
<math display="block"> a + \frac{1}{\sqrt{2}}a = \left(1 + \frac{\sqrt{2}}{2}\right)a \approx 1.707a.  </math>
Its surface area can be calculated by adding all the area of four equilateral triangles and four squares:{{r|berman}}
<math display="block"> \left(5 + \sqrt{3}\right)a^2 \approx 6.732a^2. </math>
Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them:{{r|berman}}
<math display="block"> \left(1 + \frac{\sqrt{2}}{6}\right)a^3 \approx 1.236a^3. </math>

[[File:Pirámide cuadrada elongada.stl|thumb|3D model of a elongated square pyramid.]]
The elongated square pyramid has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the equilateral square pyramid, the [[cyclic group]] <math> C_{4v} </math> 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, <math> \arccos(-1/3) \approx 109.47^\circ </math>,
* The dihedral angle of an elongated square bipyramid between two adjacent squares is the dihedral angle of a cube between those, <math> \pi/2 = 90^\circ </math>,
* The dihedral angle of an equilateral square pyramid between square and triangle is <math> \arctan \left(\sqrt{2}\right) \approx 54.74^\circ </math>. 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 <math display="block"> \arctan\left(\sqrt{2}\right) + \frac{\pi}{2} \approx 144.74^\circ. </math>