solid angle of rectangular pyramid

We calculate the apical solid angle of a rectangular pyramidMathworldPlanetmath, as an example of using the of van Oosterom and Strackee for determining the solid angle Ω subtended at the origin by a triangleMathworldPlanetmath:

tanΩ2=r1×r2r3(r1r2)r3+(r2r3)r1+(r3r1)r2+r1r2r3 (1)

Here, r1, r2, r3 are the position vectors of the vertices of the triangle and r1,r2,r3 their .

Let the apex of the pyramid be in the origin and the vertices of the base rectangleMathworldPlanetmathPlanetmath be


where a, b and h are positive numbers.  We take the half-triangle of the base determined by the three vertices


with the position vectors r1, r2, r3, respectively.  Then we have in the numerator of (1) the scalar triple productMathworldPlanetmath

r1×r2r3=|abh-abha-bh|=a|bh-bh|+b|h-aha|+h|-aba-b|= 4abh.

The vectors have the common length a2+b2+h2, and the denominator of (1) then attains the value 4h2a2+b2+h2.  Thus the formula (1) gives


which result may be reformulated by using the goniometric formula



sinΩ2=ab(a2+h2)(b2+h2). (2)

Thus the whole apical solid angle of the rectangular pyramid is

Ω= 4arcsinab(a2+h2)(b2+h2). (3)

A variant of (3) is found in [3].

In the special case of a regular pyramid we have simply

Ω= 4arcsina2a2+h2 (4)

where 2a is the side ( of the base square.

Note that in (2), the quotients aa2+h2 and bb2+h2 are sines of certain angles in the pyramid.


  • 1 A. van Oosterom & J. Strackee:  A solid angle of a plane triangle.  – IEEE Trans. Biomed. Eng. 30:2 (1983); 125–126.
  • 2 M. S. Gossman & A. J. Pahikkala & M. B. Rising & P. H. McGinley:  Providing solid angle formalism for skyshine calculations.  – Journal of Applied Clinical Medical Physics 11:4 (2010); 278–282.
  • 3 M. S. Gossman & A. J. Pahikkala & M. B. Rising & P. H. McGinley:  Letter to the editor.  – Journal of Applied Clinical Medical Physics 12:1 (2011); 242–243.
  • 4 M. S. Gossman & M. B. Rising & P. H. McGinley & A. J. Pahikkala:  Radiation skyshine from a 6 MeV medical accelerator.  – Journal of Applied Clinical Medical Physics 11:3 (2010); 259–264.
Title solid angle of rectangular pyramid
Canonical name SolidAngleOfRectangularPyramid
Date of creation 2013-03-22 19:16:02
Last modified on 2013-03-22 19:16:02
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Example
Classification msc 15A72
Classification msc 51M25
Related topic CyclometricFunctions