We establish a sharp reciprocity inequality for modulus in compact metric spaces
X
X
with finite Hausdorff measure. In particular, when
X
X
is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M. Romney [Ann. Acad. Sci. Fenn. Math. 44 (2019), pp. 681-692]. More specifically, we obtain a sharp inequality between the modulus of the family of curves connecting two disjoint continua
E
E
and
F
F
in
X
X
and the modulus of the family of surfaces of finite Hausdorff measure that separate
E
E
and
F
F
. The paper also develops approximation techniques, which may be of independent interest.