  The fuzzy disc, introduced by the authors of Ref.[1], is a disc-shaped region
in a noncommutative plane, and is a fuzzy approximation of a commutative disc.
In this paper we show that one can introduce a concept of angles to the fuzzy
disc, by using the phase operator and phase states known in quantum optics. We
gave a description of a fuzzy disc in terms of operators and their commutation
relations, and studied properties of angular projection operators. A similar
construction for a fuzzy annulus is also given. As an application, we
constructed fan-shaped soliton solutions of a scalar field theory on a fuzzy
disc, which corresponds to a fan-shaped D-brane. We also applied this concept
to the theory of noncommutative gravity that we proposed in Ref.[2]. In
addition, possible connections to black hole microstates, holography and an
experimental test of noncommutativity by laser physics are suggested.
