A certain torus has a circular vertical cross-section of radius a centred on a horizontal circle of radius c (> a).
(a) Find the volume V and surface area A of the torus, and show that they can be written as
V=\frac{\pi^2}{4}\left(r_{\mathrm{o}}^2-r_{\mathrm{i}}^2\right)\left(r_{\mathrm{o}}-r_{\mathrm{i}}\right), \quad A=\pi^2\left(r_{\mathrm{o}}^2-r_{\mathrm{i}}^2\right),
where r_{\mathrm{o}} \text { and } r_{\mathrm{i}} are, respectively, the outer and inner radii of the torus.
(b) Show that a vertical circular cylinder of radius c, coaxial with the torus, divides A in the ratio
\pi c+2 a: \pi c-2 a.