Editing Gyroelongated square pyramid (section)

== Properties ==
The surface area of a gyroelongated square pyramid with edge length <math> a </math> is:{{r|berman}}
<math display="block"> \left(1 + 3\sqrt{3}\right)a^2 \approx 6.196a^2, </math>
the area of twelve equilateral triangles and a square. Its volume:{{r|berman}}
<math display="block"> \frac{\sqrt{2} + 2\sqrt{4 + 3\sqrt{2}}}{6}a^3 \approx 1.193a^3, </math>
can be obtained by slicing the square pyramid and the square antiprism, after which adding their volumes.{{r|berman}}

It has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the square pyramid, the [[cyclic group]] <math> C_{4v} </math> of order eight. Its [[dihedral angle]] can be derived by calculating the angle of a square pyramid and square antiprism in the following:{{r|johnson}}
* the dihedral angle of an equilateral square pyramid between two adjacent triangles, approximately <math> 109.47^\circ </math>
* the dihedral angle of a square antiprism between two adjacent triangles, approximately <math> 127.55^\circ </math>, and between a triangle to its base is <math> 103.83^\circ </math>
* the dihedral angle between two adjacent triangles, on the edge where an equilateral square pyramid is attached to a square antiprism, is <math> 158.57^\circ</math>, for which by adding the dihedral angle of an equilateral square pyramid between its base and its lateral face <math> 54.74^\circ </math> and the dihedral angle of a square antiprism between two adjacent triangles.