The formulas for the volume of a sphere, or the area of a circle is something we take for granted without fully understanding why it is such. For those who read my other articles, you’ll know that I have a principle that you don’t understand something well enough, until you can derive it from basic principles. So, in this article, we will delve into why the volumes of spheres and circles are such, and how we can generalise it to hyper-spheres.
Spheres and circles
We will start gently by looking at a different approach to derive the volume and surface area formulas for spheres and circles. First, I think it is good to remind ourselves what it means for an shape to be a sphere (or hyper-sphere). Spheres are just a collection of points (in Rn) that are a distance r away from the a center point. Now consider a section of the following sphere:
Looking at the geometry, we can figure out the volume element dV representing the infinitesimal volume. This volume is simply the volume of the rectangular box at the end of the sphere, which is
Now calculating the volume is trivial, just integrate over the entire sphere
Simple isn’t it?
Now we can generalise this approach of finding the volume element and integrating to n dimensions. The most difficult part of this generalisation is finding out what the volume element will be. This requires looking at the pattern in the lower dimensions, and some clever visualisations.
Let us consider an angle between two axes, and what needs to be ‘done to it’, so to speak, to orient the length to a fixed axis, say x. In three dimensions, the first power of the sine of angles is taken to orient the r to the x axis. Similarly, we need to take higher powers of sine for the angles in higher dimensions. This takes a while to grasp, but I think imagining the orientation as steps from one axis to another helps. Given that, we can find the volume element
This formula looks daunting, especially with the Pi products, but the best way to understand it is to think about it geometrically, and step by step.
Now that we have gotten our volume element, all that is left is to integrate it. In order to understand how to go about integrating this beast, we need to introduce the Beta function.
The beta function is a special function defined as
This function doesn’t seem related to our integral at all, but consider the following substitution
Looks similar to our integral doesn’t it? Before we apply this, we need to introduce a relationship between the Beta function and the Gamma function
For those that are not familiar with the Gamma function (it is very closely related to the factorial), I suggest to read up on it and derive the above relation.
Solving the integral
Now we are finally in the position to solve the integral. Using all the above information, we write
This is now just a product of Beta functions:
Cancelling the common terms and since the Gamma function of half is the square root of pi, we finally get the volume expression
Although deriving the volume of sphere is not necessary to use it, I firmly believe that knowing where to comes from can improve our general understanding of mathematics and allow us to extend the existing principles to new premises. The volume of hyper spheres is such an example. Understanding where volume comes from has allowed us to find an elegant formula that generalises this concept to however many dimensions we like. This is the power of first principles thinking. I would strongly suggest you to apply this method of learning and investigating to improve your understanding of any quantitative subject.