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We have performed a temperature dependent angle-resolved photoemissionspectroscopy (ARPES) study of the tri-layer high-$T_c$ cuprate superconductor(HTSC) Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\delta}$ (Bi2223), and have shown thatthe \textquotedblleft effective\textquotedblright superconducting (SC) gap$\Delta_{\rm{sc}}$ defined at the end point of the Fermi arc and the $T_c$ (=110 K) approximately satisfies the weak-coupling BCS-relationship2$\Delta_{\rm{sc}}$ = 4.3$k_{\rm{B}}T_c$. Combining this result with previousARPES results on single- and double-layer cuprates, we show that therelationship between 2$\Delta_{\rm{sc}}$ = 4.3$k_{\rm{B}}T_c$ holds for variousHTSCs. Furthermore, at $T$ $\sim$ $T_c$, the quasi-patricle width at the endpoint of the Fermi arc is found to coincide with $\Delta_{\rm{sc}}$, consistentwith the context of Planckian dissipation.

An energy scale directly related to superconductivity in the high-$T_c$ cuprate superconductors: Universality from the Fermi arc picture