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 Rn:
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.
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