How many steradians in a sphere?

At first, let us define an angle in radians:

Consider an arc of lenght L over an angle alfa in a circle with radius R. The angle alfa is defined as alfa=L/R [in radians]. Similarly, an stereo angle is defined in a sphere with radius R over an area S, and the stereo angle alfa is defined as: alfa=S/R^2 [in steradians]. The sphere has S=4.pi.R^2, so the corresponding angle of the sphere in steradians is

alfa=S/R^2

alfa=4.pi.R^2/R^2

alfa=4.pi [steradians]