Chordal graphs are undirected graphs in which every cycle of length at least four has a chord. They are sometimes called rigid circuit graphs or perfect elimination graphs; the last name reflects their utility in modelling Gaussian elimination on sparse matrices. The main result of this paper is that a chordal graph with $n$ vertices and $m$ edges can be cut in half by removing $O(m)$ vertices. A similar result holds if the vertices have non-negative weights and we want to bisect the graph by weight, or even if we want to bisect the graph simultaneously by several unrelated sets of weights.