  Given a lattice $\Lambda \subset \mathbb{R}^n$, we consider its Minkowski
reduced basis and the solid angle $\Omega$ spanned by the basis vectors. Such a
basis satisfies strong near-orthogonality conditions, which allow us to bound
from above and below the measure of $\Omega$. Sharp upper and lower bounds are
derived for all rank $3$ and rank $4$ lattices so that $\Omega$ always measures
in between. Extreme cases happen when $\Lambda$ is similar to the rectangular
($\mathcal{R}$) or alternating ($\mathcal{A}$) lattice. This result settles a
question raised earlier by Fukshansky and Robins in connection to sphere
packings and kissing numbers. The proof relies on a formula by Hajja and Walker
that expresses $\Omega$ as a product of $\det(\Lambda)$ and a quadratic
integral on the unit sphere $\mathbb{S}^{n-1}$. Finally, we show that for rank
5, the alternating lattice $\mathcal{A}_{5}$ no longer possesses the smallest
measure for $\Omega$.
