Why solid angle is defined as "the two-dimensional angle in three-dimensional space"?

Question

In wikipedia the solid angle is defined as follows:

In geometry, a solid angle (symbol: Ω) is the two-dimensional angle in three-dimensional space that an object subtends at a point.

Why solid angle is a two dimensional angle?

Answer 1

An ordinary angle is a measure of the length subtended by a one-dimensional subset of the plane. For example, an angle of $60^\circ$ is subtended from a point by a $\frac16$ portion of a circle centered at that point. The $\frac16$ arc that subtends the angle is a one-dimensional subset of the two-dimensional plane.

By analogy, a solid angle is subtended by a two-dimensional subset of the three-dimensional space. A solid angle measuring 1 steradian is the solid angle subtended from a point by a certain two-dimensional subset of a sphere centered at that point.

Answer 2

On a straight line, the length of an object is the one-dimensional extent of it. On a flat plane, the area of an object is the two-dimensional extent of it. It is nonetheless given as a single value, and the unit of that value is the square of the unit of length.

Similarly, on a circle, the angle subtended by an object is the one-dimensional length of the circle "covered" by that object. On a sphere, the solid angle subtended by an object is the two-dimensional portion of the sphere "covered" by that object. It is nonetheless given as a single value.

The unit of solid angle, the steradian, is dimensionless, but it is defined such that the area covered by one steradian on the unit sphere is equal to $1$. (There are thus $4\pi$ steradians on the sphere.) This is analogous to the radian, which too is dimensionless, and is defined such that the length covered by one radian on the unit circle is equal to $1$ (so there are $2\pi$ radians in the circle).