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According to mathworld, let the sphere have radius $R$, then the surface area a spherical cap of height $h$ and base radius $a$ is given by $$S=2\pi Rh=2\pi(a^2+h^2).$$

What is this value for an n-dimensional hypersphere?
If it helps simplify the problem we can assume $R=1$ and $a=0.5$.

Many thanks.

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  • $\begingroup$ The terms "$n$-dimensional" and "hyper<object>" have different meanings to mathematicians than to the public at large. To clarify, by "$n$-dimensional hypersphere" do you mean (1) An $n$-dimensional sphere (i.e., a sphere in $\mathbf{R}^{n+1}$), or (2) A sphere in $\mathbf{R}^{n}$? $\endgroup$
    – Andrew D. Hwang
    Commented Apr 17, 2017 at 11:12
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    $\begingroup$ @AndrewD.Hwang thanks for the clarification, I wanted to mean a sphere in $R^n$ $\endgroup$
    – dontloo
    Commented Apr 17, 2017 at 12:02
  • $\begingroup$ There are easier forms for asymptotic approximations of these values, when n is large. For a sphere of radius 1, the spherical cap of height a has volume/surface area (1-a^2)^[n/2 + o(n)]. $\endgroup$
    – TMM
    Commented Apr 17, 2017 at 12:12
  • $\begingroup$ @TMM thank you that's very useful, do you have any idea about the asymptotic approximation of the ratio between the cap and the total surface area? $\endgroup$
    – dontloo
    Commented Apr 17, 2017 at 12:30
  • $\begingroup$ According to that link, the surface area is half that: $S = \pi(a^2+h^2)$. $\endgroup$
    – Brian Tung
    Commented Dec 13, 2022 at 1:31

1 Answer 1

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The solution requires special classes of functions, namely regularized incomplete beta functions $I_{\sin^2 \Phi} (\frac{n-1}{2}, \frac{1}{2})$ and Gamma functions. The result and its derivation can e.g. be found in the very nice article by S. Li here. Let $\Phi$ be the angle of the cap, so $a = R\sin \Phi$.

Then the surface area of the cap is $$ A(\Phi) = R^{n-1} \frac{\pi^{n/2}}{\Gamma(n/2)} I_{\sin^2\Phi} (\frac{n-1}{2}, \frac{1}{2}) $$

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  • $\begingroup$ Thanks for the answer, I just saw that paper as well, I think one crucial factor is the regularized incomplete beta functions, however after some googling I still don't have a clue what is the function like, especially how it grows wrt n with different $\phi$ values, could you explain a bit? $\endgroup$
    – dontloo
    Commented Apr 17, 2017 at 14:15
  • $\begingroup$ It would be great if you could also share some insight on how the ratio between the cap and the total surface area grows (asymptotically) wrt n, thank you :) $\endgroup$
    – dontloo
    Commented Apr 17, 2017 at 14:18
  • $\begingroup$ For constant angles $\Phi \in (0, \pi/2)$, the ratio of cap surface area to total area is exponentially small. $\endgroup$
    – Chris Jones
    Commented Mar 29, 2019 at 21:49

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