  We obtain an integral formula for the volume of non-toric tri-Sasaki Einstein
manifolds arising from nonabelian hyperkahler quotients. The derivation is
based on equivariant localization and generalizes existing formulas for Abelian
quotients, which lead to toric manifolds. The formula is particularly valuable
in the context of AdS$_{4}\times Y_{7}$ vacua of M-theory and their field
theory duals. As an application, we consider 3d $\mathcal N=3$ Chern-Simons
theories with affine ADE quivers. While the $\widehat A$ series corresponds to
toric $Y_{7}$, the $\widehat D$ and $\widehat E$ series are non-toric. We
compute the volumes of the corresponding seven-manifolds and compare to the
prediction from supersymmetric localization in field theory, finding perfect
agreement. This is the first test of an infinite number of non-toric
AdS$_4$/CFT$_3$ dualities.
