In this paper we investigate how the volume of a hyperbolic manifold increases under the process of removing a curve, that is, Dehn drilling. If the curve we remove is a geodesic, we show that for a certain family of manifolds the volume increase is bounded above by $\pi \cdot l$ where $l$ is the length of the geodesic drilled. We further construct examples to show that there is no lower bound to the volume increase in terms of a monotonic unbounded function of length. In particular, this shows that volume increase is not bounded below linearly in length.

1991 Mathematics Subject Classification: primary 51M10, 51M20; secondary 51M25, 52C25.