Fuzzy Geographical Entities (FGEs) refer in this paper to geographical entities with fuzzy spatial extent. The use of FGEs in geographical information systems requires the existence of operators capable of processing them. In this paper, our contribution to that field focuses on the computation of areas. Two methods are considered, one crisp due to Rosenfeld (1984), which has limited applicability, and the other fuzzy, which is a new approach. The new fuzzy area operator gives more information about the possible values of the area and enables the fuzziness in the spatial extent of the entity to be propagated to the area. Crisp and fuzzy areas have different meanings, and the use of one or the other depends not only on the purpose of the computation but also on the semantics of the membership functions. When the FGEs are represented by normal fuzzy sets, the fuzzy area operator generates fuzzy numbers, and therefore arithmetic operations can be performed with them using fuzzy arithmetic. However, we show that care must be taken with the use of the fuzzy arithmetic operators because, in some situations, the usual operators should not be applied. Properties of the Rosenfeld and fuzzy area operators are analysed, establishing a parallel with properties of the areas of crisp sets.