Editing Octahedron (section)

=== Metric properties and Cartesian coordinates ===
[[File:Octahedron.stl|thumb|3D model of regular octahedron]]
The surface area <math> A </math> of a regular octahedron can be ascertained by summing all of its eight equilateral triangles, whereas its volume <math> V </math> is twice the volume of a square pyramid; if the edge length is <math>a</math>,{{r|berman}}
<math display="block"> \begin{align}
 A &= 2\sqrt{3}a^2 &\approx 3.464a^2, \\
 V &= \frac{1}{3} \sqrt{2}a^3 &\approx 0.471a^3.
\end{align} </math>
The radius of a [[circumscribed sphere]] <math> r_u </math> (one that touches the octahedron at all vertices), the radius of an [[inscribed sphere]] <math> r_i </math> (one that tangent to each of the octahedron's faces), and the radius of a [[midsphere]] <math> r_m </math> (one that touches the middle of each edge), are:{{r|radii}}
<math display="block"> 
 r_u = \frac{\sqrt{2}}{2}a \approx 0.707a, \qquad
 r_i = \frac{\sqrt{6}}{6}a \approx 0.408a, \qquad
 r_m = \frac{1}{2}a = 0.5a.
</math>

The [[dihedral angle]] of a regular octahedron between two adjacent triangular faces is 109.47°. This can be obtained from the dihedral angle of an equilateral square pyramid: its dihedral angle between two adjacent triangular faces is the dihedral angle of an equilateral square pyramid between two adjacent triangular faces, and its dihedral angle between two adjacent triangular faces on the edge in which two equilateral square pyramids are attached is twice the dihedral angle of an equilateral square pyramid between its triangular face and its square base.{{r|johnson}}

An octahedron with edge length <math> \sqrt{2} </math> can be placed with its center at the origin and its vertices on the coordinate axes; the [[Cartesian coordinates]] of the vertices are:{{r|smith}}
<math display="block"> (\pm 1, 0, 0), \qquad (0, \pm 1, 0), \qquad (0, 0, \pm 1). </math>