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Spherical segment

Region between parallel planes intersecting a sphere

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Source updated May 1, 2025

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In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum.

The surface of the spherical segment (excluding the bases) is called spherical zone.

If the radius of the sphere is called R, the radii of the spherical segment bases are a and b, and the height of the segment (the distance from one parallel plane to the other) called h, then the volume of the spherical segment is

$V={\frac {\pi }{6}}h\left(3a^{2}+3b^{2}+h^{2}\right).$

For the special case of the top plane being tangent to the sphere, we have $b=0$ and the solid reduces to a spherical cap.

The equation above for volume of the spherical segment can be arranged to

$V={\biggl [}\pi a^{2}\left({\frac {h}{2}}{\biggr )}\right]+{\biggl [}\pi b^{2}\left({\frac {h}{2}}{\biggr )}\right]+{\biggl [}{\frac {4}{3}}\pi \left({\frac {h}{2}}\right)^{3}{\biggr ]}$

Thus, the segment volume equals the sum of three volumes: two right circular cylinders one of radius a and the second of radius b (both of height $h/2$ ) and a sphere of radius $h/2$ .

The curved surface area of the spherical zone, which excludes the top and bottom bases, is given by

$A=2\pi Rh.$

Thus the surface area of the segment depends only on the distance between the cutting planes, and not their absolute heights.

Pair of parallel planes intersecting a sphere forming a spherical segment (i.e., a spherical frustum)

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This article is adapted from the Wikipedia article “Spherical segment”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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