  We have performed a temperature dependent angle-resolved photoemission
spectroscopy (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 that
the \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-relationship
2$\Delta_{\rm{sc}}$ = 4.3$k_{\rm{B}}T_c$. Combining this result with previous
ARPES results on single- and double-layer cuprates, we show that the
relationship between 2$\Delta_{\rm{sc}}$ = 4.3$k_{\rm{B}}T_c$ holds for various
HTSCs. Furthermore, at $T$ $\sim$ $T_c$, the quasi-patricle width at the end
point of the Fermi arc is found to coincide with $\Delta_{\rm{sc}}$, consistent
with the context of Planckian dissipation.
