Surface area and volume of a sphere in finite Euclidean space

Motivation

The computation of the area and volume of a sphere is one of the classical problems of geometry. In dimension two, the corresponding formulas are already familiar:

\operatorname{Length}(\partial B^2(0,r)) = 2\pi r, \qquad \operatorname{Area}(B^2(0,r)) = \pi r^2.

In dimension three, the standard formulas are

\operatorname{Area}(\mathbb{S}^2_r)=4\pi r^2, \qquad \operatorname{Vol}(B^3(0,r))=\frac{4}{3}\pi r^3.

Historically, the three-dimensional case goes back to Archimedes. In "On the Sphere and Cylinder", Archimedes established the relation between a sphere and its circumscribed cylinder: the volume of the sphere is two-thirds the volume of the circumscribed cylinder, and the surface area of the sphere is equal to the lateral area of that cylinder.

Thus, in low dimensions, the formulas look elementary and geometric. However, once we pass from $\mathbb{R}^2$ and $\mathbb{R}^3$ to arbitrary $\mathbb{R}^n$ , the pattern is no longer obvious. One might expect powers of $\pi$ , but the exact dependence on $n$ is more subtle. The answer involves the Gamma function:

\operatorname{Vol}(B^n) = \frac{\pi^{n/2}}{\Gamma(n/2+1)}, \qquad \operatorname{Area}(\mathbb{S}^{n-1}) = \frac{2\pi^{n/2}}{\Gamma(n/2)}.

This already suggests that the general $n$ -dimensional problem is not merely a problem of elementary geometry, it belongs more naturally to analysis. The Gamma function itself arose from the problem of extending the factorial beyond integer values. This problem appeared already in the work of Wallis on quadrature, and was developed much further in the 18th century by Euler and Stirling. Euler's work on the so-called Eulerian integrals made it possible to treat expressions such as

\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}\,dt

as a continuous analogue of the factorial.

There is also a natural connection with the theory of multiple integrals. In the nineteenth century, Dirichlet studied methods for determining multiple integrals, and this analytic tradition is exactly the setting in which the volume of an $n$ -dimensional ball becomes a clean calculation. Instead of trying to visualize a high-dimensional sphere, one evaluates an integral in two different ways: first by Fubini's theorem, and then by changing to spherical coordinates.

The key example is the Gaussian integral

\int_{\mathbb{R}^n} e^{-\|x\|^2}\,dx.

On the one hand, Fubini's theorem reduces it to the one-dimensional Gaussian integral:

\int_{\mathbb{R}^n} e^{-\|x\|^2}\,dx = \left(\int_{-\infty}^{\infty}e^{-x^2}\,dx\right)^n = \pi^{n/2}.

On the other hand, spherical coordinates express the same integral as

\int_{\mathbb{R}^n} e^{-\|x\|^2}\,dx = \int_0^\infty \int_{\mathbb{S}^{n-1}} e^{-r^2} r^{n-1}\,d\sigma\,dr.

The radial part then becomes a Gamma integral. Comparing the two evaluations reveals the surface area of the unit sphere, and integrating once more in the radial variable gives the volume of the unit ball.

Therefore, the goal of this article is to derive the formulas

A_{n-1} = \operatorname{Area}(\mathbb{S}^{n-1}) = \frac{2\pi^{n/2}}{\Gamma(n/2)}

and

V_n = \operatorname{Vol}(B^n) = \frac{\pi^{n/2}}{\Gamma(n/2+1)}

Conventions

Throughout this article, we denote

B^n = B(0,1) = \{x \in \mathbb{R}^n \mid \|x\| < 1\} \quad \text{and} \quad \mathbb{S}^{n-1} = \{x \in \mathbb{R}^n \mid \|x\| = 1\}.

The volume of the unit ball is defined by the ordinary coordinate integral

V_n = \operatorname{Vol}(B^n) = \int_{B^n} dx_1\cdots dx_n.

And the surface area of the unit sphere is defined by integrating the induced area element

A_{n-1} = \operatorname{Area}(\mathbb{S}^{n-1}) = \int_{\mathbb{S}^{n-1}} d\sigma.

The goal is to compute $V_n$ and $A_{n-1}$ explicitly.

Necessary Tools

Two elementary analytic tools are used repeatedly.

\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.

Proof.

Let $I=\int_{-\infty}^{\infty}e^{-x^2}\,dx$ . Since $e^{-x^2}>0$ , $I>0$ . Then

I^2 = \left(\int_{-\infty}^\infty e^{-x^2}\,dx\right)\left(\int_{-\infty}^\infty e^{-y^2}\,dy\right) = \int_{\mathbb{R}^2} e^{-(x^2+y^2)}\,dx\,dy.

Use polar coordinates $x=r\cos\theta$ , $y=r\sin\theta$ , whose Jacobian is $r$ :

I^2 = \int_0^{2\pi}\int_0^\infty e^{-r^2}\,r\,dr\,d\theta = 2\pi\int_0^\infty r e^{-r^2}\,dr.

With $t=r^2$ and $dt=2r\,dr$ ,

I^2 = 2\pi \cdot \frac{1}{2}\int_0^\infty e^{-t}\,dt = \pi\cdot\bigl[-e^{-t}\bigr]_0^\infty = \pi.

Since $I>0$ , $I=\sqrt{\pi}$ .

■

s > 0

Proof.

1. Direct computation: $\Gamma(1) = \int_0^\infty e^{-t}\,dt = \bigl[-e^{-t}\bigr]_0^\infty = 1$ .

2. Integration by parts with $u = t^s$ and $dv = e^{-t}\,dt$ :

\Gamma(s+1) = \int_0^\infty t^s e^{-t}\,dt = \bigl[-t^s e^{-t}\bigr]_0^\infty + s\int_0^\infty t^{s-1}e^{-t}\,dt.

The boundary term vanishes since $t^s e^{-t} \to 0$ as $t \to 0^+$ and $t \to \infty$ . Therefore $\Gamma(s+1) = s\,\Gamma(s)$ .

3. From 1 and 2, induction gives $\Gamma(n+1) = n!$ , i.e., $\Gamma(n) = (n-1)!$ .

4. Substituting $t = x^2$ in the definition,

\Gamma\!\left(\tfrac{1}{2}\right) = \int_0^\infty t^{-1/2} e^{-t}\,dt = 2\int_0^\infty e^{-x^2}\,dx = \int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.

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Computing $V_n$

We will start by changing variables into polar coordinates. Consider the equation $x_1^2+\cdots+x_n^2=1$ . Since $-1\le x_1\le 1$ , write $x_1=\cos\theta_1$ . Then $x_2^2+\cdots+x_n^2=\sin^2\theta_1$ . Next write $x_2=\sin\theta_1\cos\theta_2$ , so $x_3^2+\cdots+x_n^2=\sin^2\theta_1\sin^2\theta_2$ . Iterating this substitution gives

\begin{cases} x_1 = \cos\theta_1 \\ x_2 = \sin\theta_1\cos\theta_2 \\ \quad\vdots \\ x_{n-1} = \sin\theta_1\cdots\sin\theta_{n-2}\cos\theta_{n-1} \\ x_n = \sin\theta_1\cdots\sin\theta_{n-2}\sin\theta_{n-1} \end{cases} \quad \theta_1,\dots,\theta_{n-2} \in [0,\pi),\quad \theta_{n-1} \in [0,2\pi).

Define $\sigma(\theta_1,\dots,\theta_{n-1})=(x_1,\dots,x_n)$ . Points away from the origin are written as $x=r\sigma$ . The Jacobian of $(r,\theta)\mapsto r\sigma(\theta)$ is

J(r\sigma)(r,\theta_1,\dots,\theta_{n-1}) = \det\begin{pmatrix}\dfrac{\partial x}{\partial r},\; \dfrac{\partial x}{\partial\theta_1},\;\dots,\; \dfrac{\partial x}{\partial\theta_{n-1}}\end{pmatrix} = r^{n-1}\det\begin{pmatrix}\sigma,\; \dfrac{\partial\sigma}{\partial\theta_1},\;\dots,\; \dfrac{\partial\sigma}{\partial\theta_{n-1}}\end{pmatrix}.

The remaining determinant is evaluated through its Gram matrix. Set

A=\left(\sigma,\frac{\partial\sigma}{\partial\theta_1},\dots,\frac{\partial\sigma}{\partial\theta_{n-1}}\right).

Then $|\det A|^2=\det(A^TA)$ .

Induced Metric and Area Element

The geometric origin of the induced metric is classical. For a surface parametrized by $X(u,v)\subseteq \mathbb{R}^3$ , Gauss studied the quadratic expression

ds^2 = E\,du^2+2F\,du\,dv+G\,dv^2,

where

E=\langle X_u,X_u\rangle,\qquad F=\langle X_u,X_v\rangle,\qquad G=\langle X_v,X_v\rangle.

This is the first fundamental form of the surface. In modern language, it is the metric induced on the surface by the Euclidean inner product of the ambient space. Thus the basic idea of an induced metric goes back to Gauss's theory of surfaces.

Riemann later abstracted this idea. Instead of studying only surfaces embedded in Euclidean space, he considered $n$ -dimensional manifolds equipped with an intrinsic line element. This is the origin of Riemannian geometry: a manifold is given a smoothly varying inner product on its tangent spaces, and geometric quantities such as length, angle, area, and volume are derived from this metric.

(\Sigma^m,g)

Proof.

Recall that in local coordinates $(x^i)$ ,

dx^{i_1}\wedge\cdots\wedge dx^{i_k} (\partial_{x^{j_1}},\ldots,\partial_{x^{j_k}}) = \delta^{i_1\cdots i_k}_{j_1\cdots j_k}

for all multi-indices $I=(i_1,\ldots,i_k)$ and $J=(j_1,\ldots,j_k)$ .

In positively oriented smooth coordinates, write

\omega_g = f\,dx^1\wedge\cdots\wedge dx^m.

Since $(\partial_{x^i}|_p)$ is a basis for $T_p\Sigma$ , Gram-Schmidt gives a positively oriented local orthonormal frame $(E_k)$ and a transition matrix $C=[C_{ik}]$ such that

\partial_{x^i}=\sum_{k=1}^m C_{ik}E_k.

Let $(\varepsilon^k)$ be the dual coframe of $(E_k)$ , so $\varepsilon^k(E_l)=\delta^k_l$ . Since $\omega_g$ is multilinear and alternating, and $\omega_g(E_1,\ldots,E_m)=1$ ,

f =\omega_g(\partial_{x^1},\ldots,\partial_{x^m}) =\det\bigl(\varepsilon^k(\partial_{x^i})\bigr) =\det(C).

Moreover, since $(E_k)$ is orthonormal,

\begin{aligned} g_{ij} &=\langle \partial_{x^i},\partial_{x^j}\rangle \\ &=\left\langle \sum_{k=1}^m C_{ik}E_k,\sum_{l=1}^m C_{jl}E_l\right\rangle \\ &=\sum_{k=1}^m\sum_{l=1}^m C_{ik}C_{jl}\langle E_k,E_l\rangle \\ &=\sum_{k=1}^m C_{ik}C_{jk} =(CC^T)_{ij}. \end{aligned}

Thus $G=CC^T$ , so

\det(G)=\det(CC^T)=\det(C)^2.

Since both $(\partial_{x^i})$ and $(E_k)$ are positively oriented, $\det(C)>0$ . Hence $f=\sqrt{\det(G)}$ .

■

Historically, this formula belongs to the Riemannian tradition, since the metric determines infinitesimal volume. However, the modern expression as a top-degree differential form belongs to the language of differential forms, developed systematically by Cartan and later by de Rham. The area element used above comes from the Riemannian volume form of the metric induced on the hypersurface.

Let $M^m,N^n$ be smooth manifolds and let $g$ be a Riemannian metric on $N$ . For a smooth map $F:M\to N$ , the pullback $F^*g$ on $M$ defined by

(F^*g)_p(X,Y) :=g_{F(p)}(dF_p(X),dF_p(Y)), \qquad X,Y\in T_pM.

It's not hard to prove that $F^*g$ is also a Riemannian metric on $M$ .

Let $\Sigma\subseteq M$ be a Riemannian submanifold with the metric induced from $(M^{m+1},g)$ , and let $F:U\subseteq\mathbb{R}^m\to\Sigma$ be a local parametrization. For the coordinate frame $(\partial_{x^i})$ on $U$ ,

(F^*g)_p(\partial_{x^i},\partial_{x^j}) =g_{F(p)}(dF_p(\partial_{x^i}),dF_p(\partial_{x^j})) =g_{F(p)}(\partial_{x^i}F,\partial_{x^j}F).

Therefore, if $G=[(F^*g)_{ij}]$ , the hypersurface area is computed locally by

\mathcal{A}(\Sigma) =\int_{\Sigma}\omega_g =\int_{F^{-1}(\Sigma)}\sqrt{\det(G)}\,dx^1\cdots dx^m.

The ambient metric in the present computation is Euclidean, $g=\sum_{i,j}\delta_{ij}\,dx^i\otimes dx^j$ . For the parametrization $\sigma$ of $\mathbb{S}^{n-1}$ , the induced metric matrix is

g_{ij} = \left\langle \frac{\partial\sigma}{\partial\theta_i},\, \frac{\partial\sigma}{\partial\theta_j}\right\rangle, \quad 1 \le i,j \le n-1.

This is the Gram matrix of the tangent vectors. Therefore the area element on the sphere is

d\sigma = \sqrt{|\det G|}\; d\theta_1\cdots d\theta_{n-1}.

For the graph case, if $F:U\to\Sigma$ is given by

F(x^1,\ldots,x^m)=(x^1,\ldots,x^m,f(x^1,\ldots,x^m)),

then writing $f_i=\partial_{x^i}f$ gives

\widetilde{g}_{ij} =\langle \partial_{x^i}F,\partial_{x^j}F\rangle =\delta_{ij}+f_if_j.

Equivalently,

\widetilde{G}=I_m+\nabla f^T\nabla f.

To compute the determinant, observe that every eigenvalue of $\widetilde{G}$ has the form $1+\lambda$ , where $\lambda$ is an eigenvalue of $\nabla f^T\nabla f$ . If $\lambda\neq 0$ and $v\in E_\lambda(\nabla f^T\nabla f)$ , then

(\nabla f^T\nabla f)(v) =\begin{bmatrix} f_1\langle\nabla f,v\rangle\\ f_2\langle\nabla f,v\rangle\\ \vdots\\ f_m\langle\nabla f,v\rangle \end{bmatrix} =\lambda v.

Hence $v=\frac{\langle\nabla f,v\rangle}{\lambda}\nabla f$ , so the nonzero eigenspace is spanned by $\nabla f$ . Substituting $v=\nabla f$ gives

(\nabla f^T\nabla f)(\nabla f)=|\nabla f|^2\nabla f.

Thus $|\nabla f|^2$ is the unique nonzero eigenvalue. The zero eigenspace is $(\nabla f)^\perp$ when $\nabla f\neq 0$ , and the conclusion is immediate when $\nabla f=0$ . Therefore

\operatorname{spec}(\nabla f^T\nabla f)=\{|\nabla f|^2,0,\ldots,0\},

and so

\det(\widetilde{G})=1+|\nabla f|^2.

Thus the graph area formula is

\mathcal{A}(\Sigma) =\int_{F^{-1}(\Sigma)}\sqrt{1+|\nabla f|^2}\,dx^1\cdots dx^m.

Return to the sphere. Since $|\sigma|=1$ , $\langle\sigma,\sigma\rangle=1$ . Differentiating with respect to $\theta_j$ gives

\left\langle\sigma,\, \frac{\partial\sigma}{\partial\theta_j}\right\rangle = 0, \quad \forall\, 1 \le j \le n-1.

Therefore

A^T A = \begin{bmatrix} 1 & 0 \\ 0 & G \end{bmatrix} \implies |\det A|^2 = \det(A^T A) = |\det G|,

so $|\det A|=\sqrt{|\det G|}$ . Hence the Euclidean volume element in spherical coordinates is

dx = r^{n-1}\sqrt{|\det G|}\; dr\,d\theta_1\cdots d\theta_{n-1} = r^{n-1}\,dr\,d\sigma.

Using the volume element above on the unit ball,

V_n = \int_{B^n} dx = \int_0^1\int_{\mathbb{S}^{n-1}} r^{n-1}\,d\sigma\,dr = A_{n-1}\int_0^1 r^{n-1}\,dr = \frac{A_{n-1}}{n}.

Thus evaluation of $V_n$ is reduced to evaluation of the surface area term $A_{n-1}$ .

Computing $A_{n-1}$

To evaluate $A_{n-1}$ , use the $n$ -dimensional Gaussian integral

I = \int_{\mathbb{R}^n} e^{-\|x\|^2}\,dx.

Compute $I$ in two ways.

Method 1 (Fubini).

\int_{\mathbb{R}^n} e^{-\|x\|^2}\,dx = \int_{\mathbb{R}^n} e^{-(x_1^2+\cdots+x_n^2)}\,dx_1\cdots dx_n = \left(\int_{-\infty}^\infty e^{-x^2}\,dx\right)^n = \pi^{n/2}.

Method 2 (Spherical coordinates).

I = \int_0^\infty\int_{\mathbb{S}^{n-1}} r^{n-1} e^{-r^2}\,d\sigma\,dr = A_{n-1}\int_0^\infty r^{n-1} e^{-r^2}\,dr.

With $t=r^2$ and $dt=2r\,dr$ ,

I = \frac{A_{n-1}}{2}\int_0^\infty t^{(n-2)/2} e^{-t}\,dt = \frac{A_{n-1}}{2}\,\Gamma(n/2).

Comparing both methods gives

\boxed{A_{n-1} = \frac{2\pi^{n/2}}{\Gamma(n/2)}}.

Substituting this into $V_n=A_{n-1}/n$ gives

\boxed{V_n = \frac{\pi^{n/2}}{\Gamma(n/2+1)}}.

References

[1] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., John Wiley & Sons, New York, 1999.

[2] J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 5th ed., Matrix Editions, Ithaca, 2015.

[3] J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Grad. Texts in Math., vol. 218, Springer, New York, 2013.

[4] Archimedes, On the Sphere and Cylinder, in T. L. Heath (trans.), The Works of Archimedes, Cambridge University Press, 1897.

[5] R. Roy, “The Gamma Function,” in Sources in the Development of Mathematics, Cambridge University Press, 2011.

[6] P. G. Lejeune Dirichlet, “Sur une nouvelle méthode pour la détermination des intégrales multiples,” Journal de Mathématiques Pures et Appliquées, Série 1, Tome 4, 1839, pp. 164–168.

[7] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., NIST Digital Library of Mathematical Functions, Chapter 5: Gamma Function.

[8] C. F. Gauss, Disquisitiones generales circa superficies curvas, 1827.

[9] B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, 1854. English translation: On the Hypotheses which lie at the Bases of Geometry, translated by W. K. Clifford.

[10] G. Ricci-Curbastro and T. Levi-Civita, “Méthodes de calcul différentiel absolu et leurs applications,” Mathematische Annalen, 54 (1900), 125–201.

[11] É. Cartan, Leçons sur les invariants intégraux, Hermann, Paris, 1922.

[12] G. de Rham, Variétés différentiables, Hermann, Paris, 1955.

Surface area and volume of a sphere in finite Euclidean space

Motivation

Conventions

Necessary Tools

Computing VnV_nVn​

Induced Metric and Area Element

Computing An−1A_{n-1}An−1​

References

Computing $V_n$

Computing $A_{n-1}$