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

Tags:
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?