In geometry, a ball is a region in space comprising all points within a fixed distance from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in n-dimensional Euclidean space. The volume of a unit n-ball is an important expression that occurs in formulas throughout mathematics; it generalizes the notion of the volume enclosed by a sphere in 3-dimensional space.
The n-dimensional volume of a Euclidean ball of radiusR in n-dimensional Euclidean space is:
where Γ is Euler's gamma function. The gamma function extends the factorial function to non-integerarguments. It satisfies Γ(n) = (n − 1)! if n is a positive integer and Γ(n + 1/2) = (n − 1/2) · (n − 3/2) · … · 1/2 · π1/2 if n is a non-negative integer.
Let An(R) denote the surface area of the n-sphere of radius R in (n + 1)-dimensional Euclidean space. The n-sphere is the boundary of the (n + 1)-ball of radius R, and the surface area and the volume are related by:
Thus, An(R) inherits formulas and recursion relationships from Vn + 1(R), such as
Trivially, there are also formulas in terms of factorials and double factorials.
Dimension maximizing the volume of a fixed-radius ball
Although it is generally nonsensical to compare volumes of different dimensions, such as a comparison of 3 square feet to 4 cubic feet, some will make such comparisons. Suppose that R is a fixed positive real number, and consider the volume Vn(R) as a function of the positive integer dimension n. By looking at the relationship
we see that Vn(R) ≥ Vn − 1(R) if and only if R ≥ rn where we have defined
for positive integers n. Because rn increases as n increases, it follows that n will be an integer that maximizes Vn(R) for any fixed R ∈ [rn, rn + 1].
With r0 = 0 and the first values of rn, which are r1 = 1/2 and r2 = 2/π, additional values can be computed from the above formula or via the recursion:
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.
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. 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 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.
A proof of the recursion formula relating the volume of the n-ball and an (n − 2)-ball can be given using the proportionality formula above and integration in cylindrical coordinates. Fix a plane through the center of the ball. Let r denote the distance between a point in the plane and the center of the sphere, and let θ denote the azimuth. Intersecting the n-ball with the (n − 2)-dimensional plane defined by fixing a radius and an azimuth gives an (n − 2)-ball of radius √R2 − r2. 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. The base cases of the induction are the 0-ball and the 1-ball, which can be checked directly using the facts Γ(1) = 1 and Γ(3/2) = 1/2 · Γ(1/2) = √π/2. The inductive step is similar to the above, but instead of applying proportionality to the volumes of the (n − 2)-balls, the inductive hypothesis is applied instead.
The proportionality relation can also be used to prove the recursion formula relating the volumes of an n-ball and an (n − 1)-ball. As in the proof of the proportionality formula, the volume of an n-ball can be written as an integral over the volumes of (n − 1)-balls. Instead of making a substitution, however, the proportionality relation can be applied to the volumes of the (n − 1)-balls in the integrand:
The integrand is an even function, so by symmetry the interval of integration can be restricted to [0, R]. On the interval [0, R], it is possible to apply the substitution u = (x/R)2 . This transforms the expression into
The volume of the n-ball can be computed by integrating the volume element in spherical coordinates. The spherical coordinate system has a radial coordinate r and angular coordinates φ1, …, φn − 1, where the domain of each φ except φn − 1 is [0, π), and the domain of φn − 1 is [0, 2π). The spherical volume element is:
and the volume is the integral of this quantity over r between 0 and R and all possible angles:
Each of the factors in the integrand depends on only a single variable, and therefore the iterated integral can be written as a product of integrals:
The integral over the radius is Rn/n. The intervals of integration on the angular coordinates can, by symmetry, be changed to [0, π/2]:
Each of the remaining integrals is now a particular value of the beta function:
The beta functions can be rewritten in terms of gamma functions:
This product telescopes. Combining this with the values Γ(1/2) = √π and Γ(1) = 1 and the functional equation zΓ(z) = Γ(z + 1) leads to
The volume formula can be proven directly using Gaussian integrals. Consider the function:
This function is both rotationally invariant and a product of functions of one variable each. 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 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 and thus the volumes of n-balls and areas of n-spheres can also be derived geometrically. As noted above, because a ball of radius is obtained from a unit ball by rescaling all directions in times, is proportional to , which implies . Also, because a ball is a union of concentric spheres and increasing radius by ε corresponds to a shell of thickness ε. Thus, ; equivalently, .
follows from existence of a volume-preserving bijection between the unit sphere and :
( is an n-tuple; ; we are ignoring sets of measure 0). Volume is preserved because at each point, the difference from isometry is a stretching in the xy plane (in times in the direction of constant ) that exactly matches the compression in the direction of the gradient of on (the relevant angles being equal). For , 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 Lp norm of the vector x = (x1, …, xn) in Rn is
and an Lp ball is the set of all vectors whose Lp norm is less than or equal to a fixed number called the radius of the ball. 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.
The volume of an Lp ball of radius R is
These volumes satisfy a recurrence relation similar to the one dimension recurrence for p = 2:
For p = 2, one recovers the recurrence for the volume of a Euclidean ball because 2Γ(3/2) = √π.
For most values of p, the surface area of an Lp sphere of radius R (the boundary of an Lp ball of radius R) cannot be calculated by differentiating the volume of an Lp ball with respect to its radius. 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. For p = 2 and p = ∞, this factor is one. 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.
The volume formula can be generalized even further. For positive real numbers p1, …, pn, define the unit (p1, …, pn) ball to be
The volume of this ball has been known since the time of Dirichlet:
Dimensions that are not non-negative integers
When R > 0, the defining formula
can be evaluated for any complex numbern because the reciprocal of the gamma function is an entire function. As such, one can use this formula to define volumes and surface areas for R > 0 when the number of dimensions is any complex number n, and these Vn(R) values will inherit relationships from properties of the gamma function.
In particular, Vn(R) = 0 when n is a negative even number. When n is a negative odd number, the Euler reflection formula
^Dirichlet, P. G. Lejeune (1839). "Sur une nouvelle méthode pour la détermination des intégrales multiples" [On a novel method for determining multiple integrals]. Journal de Mathématiques Pures et Appliquées. 4: 164–168.