Understanding Solid Angle and its Relation to s=rθ

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Solid angle is defined as Omega=A/R^2, where A is the area on a sphere and R is the sphere's radius, making it dimensionless like radians. The relationship between arc length and radius in angle measurement is expressed as theta=s/R, establishing that the angle is independent of the circle's size. The use of R^2 in the solid angle formula reflects the proportionality of area to the square of the radius. Both definitions emphasize the mathematical nature of angles and solid angles, rather than any underlying physics. Understanding these relationships clarifies why solid angles are measured in steradians, similar to how angles are measured in radians.
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why solid angle is A/r^2 ...why is this r^2...has it any similarity with s=rtheta??please help me
 
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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.
 
why u r using r^2..it is just to make the whole thing dimestionless or any other physics in it..??
 
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.
 
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