Question

The solid angle subtended by the total surface area of a sphere at the centre is:
A. $4\pi$
B. $2\pi$
C. $\pi$
D. $3\pi$

Answer 1

Hint:A steradian can be defined as the solid angle subtended by a unit area on its surface at the centre of a unit sphere. Any part of its surface with area $A=r^2$ subtends one steradian at its centre for a general sphere with radius $r$. A solid angle is a measure of the volume of field of view covered by a given object from a certain point. That is, from that point forward, it is a measure of how large the object looks to an observer. The point from which the object is viewed is called the solid angle apex, and it is said that the object from that point subtends its solid angle.

Complete answer:
A solid angle in steradians equals the area of a unit sphere segment in the same way as a planar angle in radians equals the length of a unit circle arc; thus the ratio of the length of a circular arc to its radius is exactly like a planar angle in radians.

The surface area $A$ of a sphere is $4\pi r^2$, it means that a sphere subtends $4\pi$ steradians at its centre. The solid angle is related to the area it cuts out of a sphere: The maximal solid angle which can be subtended at any point is $4\pi sr$, by the same statement. Hence, we can see that option A is correct.

The solid angle subtended by the total surface area of a sphere at the centre is:$4\pi$.

Note:Thus in short we can say that a solid angle is a 3D angular volume defined analogously in two dimensions to the concept of a plane angle. The steradian is the dimensionless solid angle unit, with 4π steradians in a complete sphere.