Spherical segment

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.^[1]

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.

References

^ Kern, Willis; Bland, James (1938). Solid Mensuration with Proofs (Second ed.). New York: John Wiley & Sons, Inc. pp. 97–103. Retrieved 16 May 2024.

Kern, William F.; Bland, James R. (1938). Solid Mensuration with Proofs. p. 95–97.

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[1] Kern, Willis; Bland, James (1938). Solid Mensuration with Proofs (Second ed.). New York: John Wiley & Sons, Inc. pp. 97–103. Retrieved 16 May 2024.

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