  Cool ($T\sim10^4$~K) gas is commonly observed around $z>2$ quasars as traced
by extended Ly$\alpha$ emission. These large-scale nebulae are usually studied
using circularly averaged surface brightness profiles, which suppress
information on morphological differences. Here, we revisit the Ly$\alpha$
nebulae around 78 $z\sim2-3$ quasars to obtain a novel estimate of their area
and asymmetry using a common redshift-corrected surface-brightness threshold.
We find a luminosity-area relation of the form ${{\rm log}(L_{\rm
Ly\alpha}^{\rm Neb})=a_1 log({\rm Area^{Neb})+a_0}}$. Most nebulae are
symmetric and bright, the most lopsided ones being the faintest and the less
extended. The Enormous Lyman-Alpha Nebulae, asymmetric due to the presence of
active companions, are the exceptions to this trend. By using simulations able
to reproduce $z\sim6$ quasar's nebulae, we show that the observed relation
should not vary with redshift. Finally, we discuss possible mechanisms that
drive the relation and future work needed to constrain them.
