Deriving The Volume of An N-ball - Proofs - The Volume Is Proportional To The nth Power of The Radius

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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