  We consider the ensemble of random Gaussian Laplace eigenfunctions on
$\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ (`$3d$ arithmetic random waves'), and
study the distribution of their nodal surface area. The expected area is
proportional to the square root of the eigenvalue, or `energy', of the
eigenfunction. We show that the nodal area variance obeys an asymptotic law.
The resulting asymptotic formula is closely related to the angular distribution
and correlations of lattice points lying on spheres.
