The angular momentum of a planet of mass M moving around the sun in

For small displacement ds of the planet its area can be written as

$\mathrm{dA}=\frac{1}{2} \mathrm{rd} \ell$

$=\frac{1}{2} \mathrm{rds} \sin \theta$

A.vel $=\frac{\mathrm{dA}}{\mathrm{dt}}=\frac{1}{2} \mathrm{r} \sin \theta \frac{\mathrm{ds}}{\mathrm{dt}}=\frac{\mathrm{V} \mathrm{r} \sin \theta}{2}$

$\frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=\frac{1}{2} \frac{\mathrm{mV}_{\mathrm{r}} \sin \theta}{\mathrm{m}}=\frac{\mathrm{L}}{2 \mathrm{~m}}$