We demonstrate the use of radio-frequency (rf) resonators to measure the capacitance of nanoscale 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 the measurement sensitivity values reaching down to 75zF/Hz 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 a typical sensitivity of about 40zF/Hz at room temperature with a resonator composed 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.