Last week, I wrote about the volume and outer surface area of a spherical cap using different methods, both of which gave the volume as $V = \frac{\pi}{3}R^3 (1-\cos(\alpha))^2(2-\cos(\alpha))$ and the surface area as $A_o = 2\pi R^2 (1-\cos(\alpha))$.

All very nice; however, one of my most beloved heuristics fails in this case.

If you differentiate the area of a circle ($\pi r^2$) with respect to the radius, you get $2\pi r$ – the circumference. If you differentiate the volume of a sphere ($\frac{4}{3}\pi r^3$), you get $4\pi r^2$, the surface area. In a way that can be nicely extended…

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