= Solution
For area $A$, allowed <wavevectors> have density $A/(2\pi)^2$. Include two <electron spin> states and fill a disk of radius $k_F$ at zero <temperature>. Then
$$
N=2\frac{A}{(2\pi)^2}\pi k_F^2,
\qquad n_s=N/A=\frac{k_F^2}{2\pi}.
$$
The <free electron model> has <parabolic dispersion> $E=\hbar^2k^2/(2m_e)$, with its minimum chosen as zero, so
$$
\boxed{k_F=\sqrt{2\pi n_s},\qquad E_F=\frac{\pi\hbar^2n_s}{m_e}.}
$$
These are the <Fermi wavevector> and <Fermi energy> for a spin-degenerate two-dimensional <free electron gas>. An imposed spin polarization or additional valley degeneracy would change the state count.
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