Volume of an n-ball

In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere.

The first volumes are as follows: The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:[1] where Γ is Euler's gamma function.

As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation: This allows computation of Vn(R) in approximately n / 2 steps.

The volume can also be expressed in terms of an (n − 1)-ball using the one-dimension recurrence relation: Inverting the above, the radius of an n-ball of volume V can be expressed recursively in terms of the radius of an (n − 2)- or (n − 1)-ball: Using explicit formulas for particular values of the gamma function at the integers and half-integers gives formulas for the volume of a Euclidean ball in terms of factorials.

For non-negative integer k, these are: The volume can also be expressed in terms of double factorials.

For a positive odd integer 2k + 1, the double factorial is defined by The volume of an odd-dimensional ball is There are multiple conventions for double factorials of even integers.

Under the convention in which the double factorial satisfies the volume of an n-dimensional ball is, regardless of whether n is even or odd, Instead of expressing the volume V of the ball in terms of its radius R, the formulas can be inverted to express the radius as a function of the volume: Stirling's approximation for the gamma function can be used to approximate the volume when the number of dimensions is high.

Let An − 1(R) denote the hypervolume of the (n − 1)-sphere of radius R. The (n − 1)-sphere is the (n − 1)-dimensional boundary (surface) of the n-dimensional ball of radius R, and the sphere's hypervolume and the ball's hypervolume are related by: Thus, An − 1(R) inherits formulas and recursion relationships from Vn(R), such as There are also formulas in terms of factorials and double factorials.

This is a special case of a general fact about volumes in n-dimensional space: If K is a body (measurable set) in that space and RK is the body obtained by stretching in all directions by the factor R then the volume of RK equals Rn times the volume of K. This is a direct consequence of the change of variables formula: where dx = dx1…dxn and the substitution x = Ry was made.

Another proof of the above relation, which avoids multi-dimensional integration, uses induction: The base case is n = 0, where the proportionality is obvious.

For the inductive step, assume that proportionality is true in dimension n − 1.

When the volume of the n-ball is written as an integral of volumes of (n − 1)-balls: it is possible by the inductive hypothesis 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.

The volume of the ball can therefore be written as an iterated integral of the volumes of the (n − 2)-balls over the possible radii and azimuths: The azimuthal coordinate can be immediately integrated out.

Applying the proportionality relation shows that the volume equals The integral can be evaluated by making the substitution u = 1 − (⁠r/R⁠)2 to get which is the two-dimension recursion formula.

The same technique can be used to give an inductive proof of the volume formula.

can be computed by integrating the volume element in spherical coordinates.

Using the fact that it is a product and the formula for the Gaussian integral gives: where dV is the n-dimensional volume element.

Using rotational invariance, the same integral can be computed in spherical coordinates: where Sn − 1(r) is an (n − 1)-sphere of radius r (being the surface of an n-ball of radius r) and dA is the area element (equivalently, the (n − 1)-dimensional volume element).

The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If An − 1(r) is the surface area of an (n − 1)-sphere of radius r, then: Applying this to the above integral gives the expression Substituting t = ⁠r2/2⁠: The integral on the right is the gamma function evaluated at ⁠n/2⁠.

Combining the two results shows that To derive the volume of an n-ball of radius R from this formula, integrate the surface area of a sphere of radius r for 0 ≤ r ≤ R and apply the functional equation zΓ(z) = Γ(z + 1): The relations

and thus the volumes of n-balls and areas of n-spheres can also be derived geometrically.

Volume is preserved because at each point, the difference from isometry is a stretching in the xy plane (in

, a similar argument was originally made by Archimedes in On the Sphere and Cylinder.

There are also explicit expressions for the volumes of balls in Lp norms.

The case p = 2 is the standard Euclidean distance function, but other values of p occur in diverse contexts such as information theory, coding theory, and dimensional regularization.

While the volume can be expressed as an integral over the surface areas using the coarea formula, the coarea formula contains a correction factor that accounts for how the p-norm varies from point to point.

However, if p = 1 then the correction factor is √n: the surface area of an L1 sphere of radius R in Rn is √n times the derivative of the volume of an L1 ball.

This can be seen most simply by applying the divergence theorem to the vector field F(x) = x to get For other values of p, the constant is a complicated integral.

For positive real numbers p1, …, pn, define the (p1, …, pn) ball with limit L ≥ 0 to be The volume of this ball has been known since the time of Dirichlet:[4] Using the harmonic mean

, the similarity to the volume formula for the Lp ball becomes clear.