On the relationship of gravitational constants in KK reduction

In this short note, we try to clarify a seemly trivial but often confusing question in relating a higher-dimensional physical gravitational constant to its lower-dimensional correspondence in Kaluza-Klein reduction. In particular, we re-derive the low-energy M-theory gravitational constant in terms of type IIA string coupling $g_s$ and constant ${\alpha }^{\mathrm{\prime }}$ through the metric relation between the two theories.