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Recent advancements in observational techniques have led to new tests of the general relativistic predictions for black-hole spacetimes in the strong-field regime. One of the key ingredients for several tests is a metric that allows for deviations from the Kerr solution but remains free of pathologies outside its event horizon. Existing metrics that have been used in the literature often do not satisfy the null convergence condition that is necessary to apply the strong rigidity theorem and would have allowed us to calculate the location of the event horizon by identifying it with an appropriate Killing horizon. This has led earlier calculations of event horizons of parametrically deformed metrics to either follow numerical techniques or simply search heuristically for coordinate singularities. We show that several of these metrics, almost by construction, are circular. We can, therefore, use the weak rigidity and Carter's rotosurface theorem and calculate algebraically the locations of their event horizons, without relying on expansions or numerical techniques. We apply this approach to a number of parametrically deformed metrics, calculate the locations of their event horizons, and place constraints on the deviation parameters such that the metrics remain regular outside their horizons. We show that calculating the angular velocity of the horizon and the effective gravity there offers new insights into the observational signatures of deformed metrics, such as the sizes and shapes of the predicted black-hole shadows.