| Version 3 |
Version 2 |
| \PMlinkescapeword{height} |
\PMlinkescapeword{height} |
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| Let us consider in $\mathbb{R}^3$ a ball of radius $r$ and the sphere bounding the ball. |
Let us consider in $\mathbb{R}^3$ a ball of radius $r$ and the sphere bounding the ball. |
| \begin{itemize} |
\begin{itemize} |
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\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}.
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\item Two parallel planes intersecting the ball separate between them from the ball a {\em spherical segment}, which can also be callea a {\em spherical frustum} (see the frustum).\, The curved surface of the spherical segment is the {\em spherical zone}.
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| \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 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}. |
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| \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}. |
\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} |
\end{itemize} |
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| The distance $h$ of the two planes intersecting the ball be is called the {\em height}.\, The volume of the spherical segment (and the spherical cap) is obtained from |
The distance $h$ of the two planes intersecting the ball be is called the {\em height}.\, The volume of the spherical segment (and the spherical cap) is obtained from |
| $$V \,=\, \pi h^2\left(r\!-\!\frac{h}{3}\right)$$ |
$$V \,=\, \pi h^2\left(r\!-\!\frac{h}{3}\right)$$ |
| and the area of the corresponding spherical zone (and the spherical calotte) from |
and the area of the corresponding spherical zone (and the spherical calotte) from |
| $$A \,=\, 2\pi rh.$$ |
$$A \,=\, 2\pi rh.$$ |
| The volume of a spherical sector may be calculated from |
The volume of a spherical sector may be calculated from |
| $$V \,=\, \frac{2}{3}\pi r^2h,$$ |
$$V \,=\, \frac{2}{3}\pi r^2h,$$ |
| where $h$ is the height of the spherical cap of the spherical sector. |
where $h$ is the height of the spherical cap of the spherical sector. |
