The present invention includes methods and apparatus (10) for forming a tire component upon an axially tapered building surface (100). In particular embodiments, such methods include the step of providing a mechanical system, the system including a plurality of cutting members (42) and an application surface (64) for transferring material to the axially tapered building surface, the axially tapered surface comprising a surface of revolution that varies radially along a rotational axis of the surface. Particular embodiments may also include translating a sheet of material along a translation path through the mechanical system, and cutting a strip of material from the sheet with one or more of the plurality of cutting members during the step of translating. Particular embodiments may further include mechanically transferring the strip of material from the application surface to the axially tapered building surface, the axially tapered building surface rotating about an axis of rotation during the step of translating.