Question

Why solid angle is dimensionless physical quantity ?​

Answer 1

Answer:

Solid angle is the surface area swept by a circle of radius r, and it is the ratio between the surface area and the radius of the sphere. It has units similar to that of the angle, i.e. radians or degree. ... Since it is the ratio of speed, it has no dimension or units.

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Explanation:

Answer 2

Answer :

The solid angle is defined as the area on the unit sphere subtended by the angle divided by one unit area. It's a ratio so it's a single dimensionless number.

I see why you think it should be a 2D quantity, because the surface of a sphere, and any patch on it, is a 2D manifold and you need two quantities (traditionally θ and ϕ) to map it. When you calculate an area on the sphere you are basically calculating a definite integral over θ and ϕ, and the result is of course just a single number. You do lose information in the process - for example you just know the total area not the shape of the patch on the sphere.

The solid angle that covers the whole sphere is of course 4π/1 or 4π

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