calculating the solid angle of disc

We determine the solid angle formed by a disc when one is looking at it on the normal line of its plane set to the center of it.

Let us look the disc from the origin and let the disc with radius R situate such that its plane is parallelMathworldPlanetmathPlanetmathPlanetmath to the xy-plane and the center is on the z-axis at  (0, 0,h)  with  h>0.  Into the

Ω=-ada1r=adar|r|3 (1)

of the parent entry (, we may substitute the position vectorr=xi+yj+hk  of the directed surface element  da=kda, getting


Now we can use a annulusMathworldPlanetmath ( surface element  da=2πϱdϱ  where  ϱ2=x2+y2, whence the surface integral may be calculated as


Thus we have the result

Title calculating the solid angle of disc
Canonical name CalculatingTheSolidAngleOfDisc
Date of creation 2013-03-22 18:19:36
Last modified on 2013-03-22 18:19:36
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Example
Classification msc 15A72
Classification msc 51M25
Related topic SubstitutionNotation
Related topic AngleOfViewOfALineSegment