**The Volume Is Proportional To The nth Power of The Radius**

An important step in several proofs about volumes of *n*-balls, and a generally useful fact besides, is that the volume of the *n*-ball of radius *R* is proportional to *R**n*:

The proportionality constant is the volume of the unit ball.

The above relation has a simple inductive proof. The base case is *n* = 0, where the proportionality is obvious. For the inductive case, assume that proportionality is true in dimension *n* − 1. Note that the intersection of an *n*-ball with a hyperplane is an (*n* − 1)-ball. When the volume of the *n*-ball is written as an integral of volumes of (*n* − 1)-balls:

it is possible by the inductive assumption to remove a factor of *R* from the radius of the *n* − 1 ball to get:

Making the change of variables *t* = *x*/*R* leads to:

which demonstrates the proportionality relation in dimension *n*. By induction, the proportionality relation is true in all dimensions.

Read more about this topic: Deriving The Volume Of An N-ball, Proofs

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