I Prove that solid angle of any closed surface is 4pi

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1. Jul 18, 2017

I googled a lot on proof of Gauss theorem and nearly every other proof (on web and so on books) state that solid angle of closed surface is 4pi but I can't find the proof of this nowhere !

I tried setting up the integral but don't know how to proceed furthur :
Ω=∫(cosθ/r^2)*dA

Also The one more thing which confuses me is http://mathworld.wolfram.com/SolidAngle.html proves solid angle of some other 3d shapes , those shapes also have close surfaces then why their solid angle is not equal to 4pi sr?

2. Jul 18, 2017

Nidum

I think that these calculations are for the solid angle subtended by just one facet on the 3D shapes .

Add the solid angles subtended by all the facets on one of the shapes together and it comes to 4pi .

3. Jul 18, 2017

@Nidum : Uh my bad, but how shall one prove that for any closed surface solid angle is always 4pi sr ?

Last edited: Jul 18, 2017
4. Jul 18, 2017

Nidum

The mathematicians can help you with formal proof but intuitively you can see why it is true by considering the arbitrary shape to be inside a sphere and the central point of the sphere to be inside the arbitrary shape . I'll leave you to think about that .

5. Jul 20, 2017

olivermsun

By solid angle of closed surface, do you mean the solid angle subtended/covered by a surface, similar to the arc length subtended by a (plane) angle?