Editing Solid angle (section)

===Pyramid===
The solid angle of a four-sided right rectangular [[Pyramid (geometry)|pyramid]] with [[apex (geometry)|apex]] angles {{mvar|a}} and {{mvar|b}} ([[dihedral angle]]s measured to the opposite side faces of the pyramid) is
<math display=block>\Omega = 4 \arcsin \left( \sin \left({a \over 2}\right) \sin \left({b \over 2}\right) \right). </math>

If both the side lengths ({{math|''α''}} and {{math|''β''}}) of the base of the pyramid and the distance ({{math|''d''}}) from the center of the base rectangle to the apex of the pyramid (the center of the sphere) are known, then the above equation can be manipulated to give

<math display=block>\Omega = 4 \arctan \frac {\alpha\beta} {2d\sqrt{4d^2 + \alpha^2 + \beta^2}}. </math>

The solid angle of a right {{mvar|n}}-gonal pyramid, where the pyramid base is a regular {{mvar|n}}-sided polygon of circumradius {{mvar|r}}, with a 
pyramid height {{mvar|h}} is

<math display=block>\Omega = 2\pi - 2n \arctan\left(\frac {\tan \left({\pi\over n}\right)}{\sqrt{1 + {r^2 \over h^2}}} \right). </math>

The solid angle of an arbitrary pyramid with an {{math|''n''}}-sided base defined by the sequence of unit vectors representing edges {{math|{''s''<sub>1</sub>, ''s''<sub>2</sub>}, ... ''s''<sub>''n''</sub>}} can be efficiently computed by:<ref name ="Mazonka"/>

<math display=block> \Omega = 2\pi - \arg \prod_{j=1}^{n} \left(
    \left( s_{j-1} s_j \right)\left( s_{j} s_{j+1} \right) -
    \left( s_{j-1} s_{j+1} \right) +
    i\left[ s_{j-1} s_j s_{j+1} \right]
  \right).
</math>

where parentheses (* *) is a [[scalar product]] and square brackets [* * *] is a [[scalar triple product]], and {{mvar|i}} is an [[imaginary unit]]. Indices are cycled: {{math|''s''<sub>0</sub> {{=}} ''s''<sub>''n''</sub>}} and {{math|''s''<sub>1</sub> {{=}} ''s''<sub>''n'' + 1</sub>}}.  The complex products add the phase associated with each vertex angle of the polygon.  However, a multiple of
<math>2\pi</math> is lost in the branch cut of <math>\arg</math> and must be kept track of separately.  Also, the running product of complex phases must scaled occasionally to avoid underflow in the limit of nearly parallel segments.