  We determine the large cardinal consistency strength of the existence of a
$\lambda$-supercompact cardinal $\kappa$ such that GCH fails at $\lambda$.
Indeed, we show that the existence of a $\lambda$-supercompact cardinal
$\kappa$ such that $2^\lambda \geq \theta$ is equiconsistent with the existence
of a $\lambda$-supercompact cardinal that is also $\theta$-tall. We also prove
some basic facts about the large cardinal notion of tallness with closure.
