Understanding Solid Angle and its Relation to s=rθ

[hasanhabibul]

why solid angle is A/r^2 ...why is this r^2...has it any similarity with s=rtheta??please help me

 

[Meir Achuz]

They are similar.
Angle is defined as theta=s/R, where s is an arc length and R the radius of a circle.
Solid angle is defined as Omega=A/R^2, where A is an area on a sphere and R is the radius of the sphere.
In each case, the angle and solid angle are dimensionless, but given the names radian and steradian for convenience.
If the surface is not on a sphere, then differential vectors must be used in the definition of solid angle.

 

[hasanhabibul]

why u r using r^2..it is just to make the whole thing dimestionless or any other physics in it..??

 

[Meir Achuz]

It does make solid angle dimensionless, but there are other reasons too.
Why divide s by R for radians? Because the arc length is proportional to R.
Dividing the arc length by R makes the angle measure in radians independent of the size of the circle. The same reasoning gives R^2 for steradians, because the area isl proportional to R^2. This is all simple mathematics, independent of any physics.