Distributions sensitive to the underlying event are studied in events containing one or more charged-particle jets produced in $pp$ collisions at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$ with the ATLAS detector at the Large Hadron Collider (LHC). These measurements reflect $800\text{ }\text{ }\ensuremath{\mu}{\mathrm{b}}^{\ensuremath{-}1}$ of data taken during 2010. Jets are reconstructed using the anti-${k}_{t}$ algorithm with radius parameter $R$ varying between 0.2 and 1.0. Distributions of the charged-particle multiplicity, the scalar sum of the transverse momentum of charged particles, and the average charged-particle ${p}_{\mathrm{T}}$ are measured as functions of ${p}_{\mathrm{T}}^{\mathrm{jet}}$ in regions transverse to and opposite the leading jet for $4\text{ }\text{ }\mathrm{GeV}<{p}_{\mathrm{T}}^{\mathrm{jet}}<100\text{ }\text{ }\mathrm{GeV}$. In addition, the $R$ dependence of the mean values of these observables is studied. In the transverse region, both the multiplicity and the scalar sum of the transverse momentum at fixed ${p}_{\mathrm{T}}^{\mathrm{jet}}$ vary significantly with $R$, while the average charged-particle transverse momentum has a minimal dependence on $R$. Predictions from several Monte Carlo tunes have been compared to the data; the predictions from Pythia 6, based on tunes that have been determined using LHC data, show reasonable agreement with the data, including the dependence on $R$. Comparisons with other generators indicate that additional tuning of soft-QCD parameters is necessary for these generators. The measurements presented here provide a testing ground for further development of the Monte Carlo models.