The Horn Torus

The Horn Torus is a solid that if formed by the revolution of a circle around a point on its circumference. Suppose the starting circle has a diameter of 1 unit and thus a circumference of $\pi$ units. Consider a small rotation of length $\delta x$ along the outer circumference that produces a wedge-shaped solid that is equivalent to a cylinder of curved surface area $\pi \delta x/2$. The total surface area of the resultant torus is given by:$$\begin{array}{rl}\text{Surface Area }& =\underset{\delta x\to 0}{lim}\sum _0^{2\pi }\pi \delta x/2\\ & ={\int }_0^{2\pi }\pi /2dx\\ & =[\pi x/2{]}_0^{2\pi }\\ & ={\pi }^2\end{array}$$This is beautifully simple. If we envisage $\pi$ as the area of a circle with unit radius then ${\pi }^2$ can be envisaged as the surface area of a horn torus with unit tube diameter. 

While we're here, we may as well calculate the volume of a horn torus with a unit tube diameter. The calculation of volume is very similar to that of the surface area. Consider a small rotation of length $\delta x$ along the outer circumference that produces a wedge-shaped solid that is equivalent to a cylinder of cross-sectional area $\pi /4$ and thickness $\delta x/2$. The volume of this wedge shape is $\pi \delta x/8$. The total volume of the resultant torus is given by:$$\begin{array}{rl}\text{Volume }& =\underset{\delta x\to 0}{lim}\sum _0^{2\pi }\pi \delta x/8\\ & ={\int }_0^{2\pi }\pi /8dx\\ & =[\pi x/8{]}_0^{2\pi }\\ & ={\pi }^2/4\end{array}$$