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Spherical Cap

\begin{figure}\begin{center}\BoxedEPSF{SphericalCap.epsf scaled 1000}\end{center}\end{figure}

A spherical cap is the region of a Sphere which lies above (or below) a given Plane. If the Plane passes through the Center of the Sphere, the cap is a Hemisphere. Let the Sphere have Radius $R$, then the Volume of a spherical cap of height $h$ and base Radius $a$ is given by the equation of a Spherical Segment (which is a spherical cut by a second Plane)

V_{\rm spherical\ segment}={\textstyle{1\over 6}}\pi h(3a^2+3b^2+h^2)
\end{displaymath} (1)

with $b=0$, giving
V_{\rm cap}={\textstyle{1\over 6}}\pi h(3a^2+h^2).
\end{displaymath} (2)

Using the Pythagorean Theorem gives
\end{displaymath} (3)

which can be solved for $a^2$ as
\end{displaymath} (4)

and plugging this in gives the equivalent formula
V_{\rm cap}={\textstyle{1\over 3}}\pi h^2(3R-h).
\end{displaymath} (5)

In terms of the so-called Contact Angle (the angle between the normal to the sphere at the bottom of the cap and the base plane)
\end{displaymath} (6)

\alpha\equiv \sin^{-1}\left({R-h\over R}\right),
\end{displaymath} (7)

V_{\rm cap}= {\textstyle{1\over 3}}\pi R^3(2-3\sin\alpha+\sin^3\alpha).
\end{displaymath} (8)

Consider a cylindrical box enclosing the cap so that the top of the box is tangent to the top of the Sphere. Then the enclosing box has Volume

$\displaystyle V_{\rm box}$ $\textstyle =$ $\displaystyle \pi a^2h=\pi(R\cos\alpha)[R(1-\sin\alpha)]$  
  $\textstyle =$ $\displaystyle \pi R^3(1-\sin\alpha-\sin^2\alpha+\sin^3\alpha),$ (9)

so the hollow volume between the cap and box is given by
V_{\rm box}-V_{\rm cap} = {\textstyle{1\over 3}}\pi R^3(1-3\sin^2\alpha+2\sin^3\alpha).
\end{displaymath} (10)

If a second Plane cuts the cap, the resulting Spherical Frustum is called a Spherical Segment. The Surface Area of the spherical cap is given by the same equation as for a general Zone:

S_{\rm cap}=2\pi Rh.
\end{displaymath} (11)

See also Contact Angle, Dome, Frustum, Hemisphere, Solid of Revolution, Sphere, Spherical Segment, Torispherical Dome, Zone

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© 1996-9 Eric W. Weisstein