The analysis of shells damped by a constrained viscoelastic layer is extended to toroidal segments. The absorber consists of thin curved strips bonded to the torus with a thin viscoelastic layer. When applied to the torus, the absorber is most effective on response modes with two or more circumferential waves. In this way it resembles the damped cylinder. However, for a given weight of absorber, the torus is damped more than the cylinder. The reason is that the torus induces and couples those circumstantial harmonics that are damped most effectively by this absorber.