The powerful Fanaroff-Riley class II (FR II) radio galaxy Cygnus A exhibits primary and secondary hotspots in each lobe. A 2 Ms Chandra X-ray image of Cygnus A has revealed an approximately circular hole, with a radius of 3.9 kpc, centered on the primary hotspot in the eastern radio lobe, hotspot E. We infer the distribution of X-ray emission on our line of sight from an X-ray surface brightness profile of the radio lobe adjacent to the hole and use it to argue that the hole is excavated from the radio lobe. The surface brightness profile of the hole implies a depth at least 1.7 ± 0.3 times greater than its projected width, requiring a minimum depth of 13.3 ± 2.3 kpc. A similar hole observed in the 5 GHz Very Large Array radio map reinforces the argument for a cavity lying within the lobe. We argue that the jet encounters the shock compressed intracluster medium at hotspot E, passing through one or more shocks as it is deflected back into the radio lobe. The orientation of Cygnus A allows the outflow from hotspot E to travel almost directly away from us, creating an elongated cavity, as observed. These results favor models for multiple hotspots in which an FR II jet is deflected at a primary hotspot, then travels onward to deposit the bulk of its power at a secondary hotspot, rather than the dentist drill model.