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.

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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

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

For the special case of the top plane being tangent to the sphere, we have ${\displaystyle b=0}$ and the solid reduces to a spherical cap.^[1]

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

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

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 ${\displaystyle h/2}$) and a sphere of radius ${\displaystyle h/2}$.

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

${\displaystyle A=2\pi Rh.}$

• Spherical cap
• Spherical wedge
• Spherical sector