Summary: A precise formulation of the hoop conjecture for four-dimensional spacetimes proposes that the Birkhoff invariant \(\beta\) for an apparent horizon in a spacetime with mass \(M\) should satisfy \(\beta \leqslant 4 \pi M\). The invariant \(\beta\) is the least maximal length of any sweepout of the 2-sphere apparent horizon by circles. An analogous conjecture in five spacetime dimensions was recently formulated, asserting that the Birkhoff invariant \(\beta\) for \(\mathrm S^1 \times \mathrm S^1\) sweepouts of the apparent horizon should satisfy \(\beta \leqslant \frac{16}{3} \pi M\). Although this hoop inequality was formulated for conventional five-dimensional black holes with 3-sphere horizons, we show here that it is also obeyed by a wide variety of black rings, where the horizon instead has \(\mathrm S^2 \times \mathrm S^1\) topology.