How to calculate the area covered by any spherical rectangle? Is there any analytic or generalized formula to calculate area covered by any rectangle having length $l$ & width $b$ each as a great circle arc on a spherical surface with a radius $R$? i.e. How to find the area $A$ of rectangle in terms of length $l$, width $b$ and radius $R$
($A=f(l, b,R)$)?

Note: Spherical rectangle is a quadrilateral having equal opposite sides but non-parallel & all the interior angles are equal in magnitude & each one is greater than $90^\circ$.


Assume we are working on a sphere of radius $1$, or consider the lengths in radians and the areas in steradians.
Extend the sides of length $l$ until they meet. This results in a triangle with sides 
$$
w,\quad\frac\pi2-\frac l2,\quad\frac\pi2-\frac l2
$$

The Spherical Law of Cosines says that
$$
\begin{align}
\cos(A)
&=\frac{\cos\left(\frac\pi2-\frac l2\right)-\cos\left(\frac\pi2-\frac l2\right)\cos(w)}{\sin\left(\frac\pi2-\frac l2\right)\sin(w)}\\
&=\frac{\sin\left(\frac l2\right)}{\cos\left(\frac l2\right)}\frac{1-\cos(w)}{\sin(w)}\\[6pt]
&=\tan\left(\frac l2\right)\tan\left(\vphantom{\frac l2}\frac w2\right)
\end{align}
$$
One quarter of the spherical excess of the rectangle is $D-\frac\pi2$ and
$$
\sin\left(D-\frac\pi2\right)=\tan\left(\frac l2\right)\tan\left(\vphantom{\frac l2}\frac w2\right)
$$
Therefore, the area of the rectangle is
$$
\bbox[5px,border:2px solid #C0A000]{4\sin^{-1}\left(\tan\left(\frac l2\right)\tan\left(\vphantom{\frac l2}\frac w2\right)\right)}
$$
Note that for small $l$ and $w$, this is approximately $lw$; and if $l+w=\pi$ (that is, the rectangle is a great circle), we get an area of $2\pi$ (one half the sphere).
Scaling for a sphere of radius $R$ gives
$$
\bbox[5px,border:2px solid #C0A000]{4R^2\sin^{-1}\left(\tan\left(\frac l{2R}\right)\tan\left(\vphantom{\frac l2}\frac w{2R}\right)\right)}
$$

Note the similarity to the formula for the area of a spherical right triangle with legs $a$ and $b$:
$$
2\tan^{-1}\left(\tan\left(\vphantom{\frac b2}\frac a2\right)\tan\left(\frac b2\right)\right)
$$
or for a sphere of radius $R$,
$$
2R^2\tan^{-1}\left(\tan\left(\vphantom{\frac b2}\frac a{2R}\right)\tan\left(\frac b{2R}\right)\right)
$$
