Torus Formulas

I've had a week to reflect on the torus. Most of my reflections have been in the past few minutes leading up to my current act of typing, but I like to think my subconscious has been mulling it over.

Remember that last week we learned that the plural of torus is tori. True, even though my spell checker has put a squiggly red line under it.

I did think of another application. Remember a torus is a donut shape. Inner tubes of tires are also tori. So the square inch amount of rubber in the tire can be found with the surface area formula and the air in the tire would be found with the volume formula.

Also, there are apparently applications I would never have come up with.

In our model of cosmometry, the torus is the fundamental form of balanced energy flow found in sustainable systems at all scales. It is the primary component that enables a seamless fractal embedding of energy flow from micro-atomic to macro-galactic wherein each individual entity has its unique identity while also being connected with all else. 

This is from the website http://cosmometry.net/the-torus---dynamic-flow-process. I do not know what this all means. I'm not even sure what all the individual words mean, but it certainly sounds very important.

Last time I listed the formulas for the torus. I found there are other ways to find volume and surface area. Instead of using the variables in the way we used last week, these formulas use r, the distance from the center of the torus to the inner edge and R the distance from the center to the outer edge. 

V =  1/4(pi)^2(r+R)(R-r)^2

S.A. = (pi)^2(R^2-r^2)

(Again, I apologize for my inability to write exponents any other way.)

Beside finding volumes and surface areas of donuts, inner tubes, and various balanced energy flows (?) there is another nice application here. To find the ratio of surface area to volume of any three-dimensional shape is an important concept. To do so with the above formulas is especially cool as it simplifies down a lot.