Spherical segment

Short description: Region between parallel planes intersecting a sphere

A spherical segment

File:01sphere2planes.pdf

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

File:02sph.seg.3D.pdf

Terminology for spherical segments.

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.

File:Geometric parameters for spherical segment.pdf

Geometric parameters for spherical segment.

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

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

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

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

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.

See also

• Spherical cap
• Spherical wedge
• Spherical sector

References

• Kern, William F.; Bland, James R. (1938). Solid Mensuration with Proofs. p. 95–97. https://archive.org/details/in.ernet.dli.2015.205959. 

External links

• Weisstein, Eric W.. "Spherical segment". http://mathworld.wolfram.com/SphericalSegment.html. 
• Weisstein, Eric W.. "Spherical zone". http://mathworld.wolfram.com/Zone.html. 
• Summary of spherical formulas