parts of a ball

Let us consider in $\mathbb{R}^{3}$ a ball of radius $r$ and the sphere bounding the ball.

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Two parallel planes intersecting the ball separate between them from the ball a spherical segment, which can also be called a spherical frustum (see the frustum). The curved surface of the spherical segment is the spherical zone.
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In the special case that one of the planes is a tangent plane of the sphere, the spherical segment is a spherical cap and the spherical zone is a spherical calotte.
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The lateral surface of a circular cone with its apex in the centre (http://planetmath.org/Sphere) of the ball divides the ball into two spherical sectors.

The distance $h$ of the two planes intersecting the ball be is called the 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^{2}h,

where $h$ is the height of the spherical cap of the spherical sector.