Explain supplementary quantities and their units in the $SI$ system.

(N/A) There are two supplementary quantities in the $SI$ system:
$(1)$ Plane angle $d \theta$
$(2)$ Solid angle $d \Omega$
$(1)$ Plane angle $d \theta$: The ratio of the arc length of a circle to its radius is called the plane angle $(d \theta)$.
From the figure,the plane angle $d \theta = \frac{\text{arc}}{\text{radius}} = \frac{ds}{r}$.
The plane angle subtended at the center by an arc of a circle having a length equal to the radius is called $1$ radian. It is represented as $rad$. The maximum value of the plane angle is $2 \pi \ rad$.
If $ds = r$,then $\theta = 1 \ rad$.
$[1^{\circ} = \frac{\pi}{180} \ rad]$ and $[1 \ rad = \frac{180}{\pi} \ \text{degree}]$.
$(2)$ Solid angle $d \Omega$: The angle subtended by an area $(\Delta A)$ on a spherical surface at the center of the sphere is called the solid angle $d \Omega$.
$d \Omega = \frac{dA}{r^2} \ \text{steradian}$.
From the figure,the solid angle $d \Omega = \frac{\text{area}}{(\text{radius})^2} = \frac{\Delta A}{r^2}$.
The maximum value of the solid angle is $4 \pi \ sr$.
The angle subtended by an area of $1 \ m^2$ on a sphere of $1 \ m$ radius at its center is called $1$ steradian. Its symbol is $sr$.
If $\Delta A = 1 \ m^2$ and $r = 1 \ m$,then $\Omega = 1 \ sr$.