  We demonstrate the use of radio-frequency (rf) resonators to measure the
capacitance of nano-scale semiconducting devices in field-effect transistor
configurations. The rf resonator is attached to the gate or the lead of the
device. Consequently, tuning the carrier density in the conducting channel of
the device affects the resonance frequency, quantitatively reflecting its
capacitance. We test the measurement method on InSb and InAs nanowires at
dilution-refrigerator temperatures. The measured capacitances are consistent
with those inferred from the periodicity of the Coulomb blockade of quantum
dots realized in the same devices. In an implementation of the resonator using
an off-chip superconducting spiral inductor we find sensitivity values reaching
down to 75~zF/$\sqHz$ at 1~kHz measurement bandwidth, and noise down to 0.45~aF
at 1~Hz bandwidth. We estimate the sensitivity of the method for a number of
other implementations. In particular we predict typical sensitivity of about
40~zF/$\sqHz$ at room temperature with a resonator comprised of off-the-shelf
components. Of several proposed applications, we demonstrate two: the
capacitance measurement of several identical 80~nm-wide gates with a single
resonator, and the field-effect mobility measurement of an individual nanowire
with the gate capacitance measured in-situ.
