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By analyzing $6.32~\mathrm{fb}^{-1}$ of $e^+e^-$ annihilation data collectedat the center-of-mass energies between 4.178 and 4.226\,GeV with the BESIIIdetector, we determine the branching fraction of the leptonic decay$D_s^+\to\tau^+\nu_\tau$ with $\tau^+\to\pi^+\pi^0\bar \nu_\tau$, to be$\mathcal{B}_{D_s^+\to\tau^+\nu_\tau}=(5.29\pm0.25_{\rm stat}\pm0.20_{\rmsyst})\%$. We estimate the product of the Cabibbo-Kobayashi-Maskawa matrixelement $|V_{cs}|$ and the $D_s^+$ decay constant $f_{D^+_s}$ to be$f_{D_s^+}|V_{cs}|=(244.8\pm5.8_{\rm stat}\pm4.8_{\rm syst})~\mathrm{MeV}$using the known values of the $\tau^+$ and $D_s^+$ masses as well as the$D_s^+$ lifetime, together with our branching fraction measurement. Combiningwith the value of $|V_{cs}|$ obtained from a global fit in the standard modeland $f_{D_s^+}$ from lattice quantum chromodynamics, we obtain$f_{D_s^+}=(251.6\pm5.9_{\rm stat}\pm4.9_{\rm syst})$\,MeV and $|V_{cs}| =0.980\pm0.023_{\rm stat}\pm0.019_{\rm syst}$.

Measurement of the branching fraction of leptonic decay $D_s^+\to\tau^+\nu_\tau$ via $\tau^+\to\pi^+\pi^0\bar \nu_\tau$