I'm told that:
the surface Area of the unit sphere (r=1) in Q-dimensions is:
$$S_Q = \frac{2\pi^{Q/2}}{\Gamma(\frac{q}{2})}$$
and the volume of the unit sphere (r=1) in Q-dimensions is:
$$V_Q = \int_{0}^{1} S_Q~ r^{(Q-1)}~dr$$
$$V_Q = \frac{S_Q}{Q}$$
I'm trying to understand the formula for the general area, and general volume of Q-dimensional Sphere by multiply the unit volume ($V_Q$), and unit surface area ($S_Q$) by a function of radius. Specifically:
$$\tilde{S}_Q(r) = S_Q ~r^{Q-1}$$ $$\tilde{V}_Q(r) = V_Q ~r^Q$$
For instance, Why does multiplying the unit surface area $S_Q$ by $r^{Q-1}$ equal the Surface area for any sphere of radius r and dimension Q?
And, why does multiplying the unit volume by $r^Q$ equal the volume for any sphere of radius r and dimension Q?