An expanding reamer for reaming or cutting a concave surface, for example, for reaming an acetabulum in preparation for implanting a prosthetic component, such as an acetabular cup or socket, during a hip arthroplasty. The reamer includes a rotating shaft cooperating with a surgical drill or other power source at one end and rotating a reamer head at the other end, and a system adapted to expand one or more blades on the reamer head. In a preferred version, the reamer head comprises a plurality of generally circular, preferably substantially flat and parallel blades, the outer blades of which are radially expandable as segments of a cutting sphere to enlarge the effective diameter of the reamer head. A transverse blade may guide expansion of the blades to move upwards as well as outward to maintain a nearly perfect cutting sphere across a range of diameters. Upon rotation of the reamer head, the blades form a portion of an effective cutting sphere that is preferably greater-than-180-degrees in order to allow greater flexibility in placement of the shaft of the reamer relative to the surface being reamed, for example, relative to the center of axis of the acetabulum.