parts of a ballPartsOfABall2008-08-10 10:01:232008-08-19 12:23:15Definitionintersection of sphere and planespherical segmentspherical frustumspherical capspherical calottespherical sector% this is the default PlanetMath preamble. as your knowledge
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Let us consider in $\mathbb{R}^3$ a ball of radius $r$ and the sphere bounding the ball.
\begin{itemize}
\item Two parallel planes intersecting the ball separate between them from the ball a {\em spherical segment}, which can also be called a {\em spherical frustum} (see the frustum).\, The curved surface of the spherical segment is the {\em spherical zone}.
\item In the special case that one of the planes is a tangent plane of the sphere, the spherical segment is a {\em spherical cap} and the spherical zone is a {\em spherical calotte}.
\item The lateral surface of a circular cone with its apex in the \PMlinkname{centre}{Sphere} of the ball divides the ball into two {\em spherical sectors}.
\end{itemize}
The distance $h$ of the two planes intersecting the ball be is called the {\em height}.
The volume of the spherical cap is obtained from
$$V \,=\, \pi h^2\left(r\!-\!\frac{h}{3}\right)$$
and the area of the corresponding spherical calotte and also a spherical zone from
$$A \,=\, 2\pi rh.$$
The volume of a spherical segment can be got as the difference of the volumes of two spherical caps.
The volume of a spherical sector may be calculated from
$$V \,=\, \frac{2}{3}\pi r^2h,$$
where $h$ is the height of the spherical cap of the spherical sector.