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 = π 6 h ( 3 a 2 + 3 b 2 + h 2 ) . {\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 b = 0 {\displaystyle b=0} and the solid reduces to a spherical cap. The equation above for volume of the spherical segment can be arranged to V = [ π a 2 ( h 2 ) ] + [ π b 2 ( h 2 ) ] + [ 4 3 π ( h 2 ) 3 ] {\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 h / 2 {\displaystyle h/2} ) and a sphere of radius h / 2 {\displaystyle h/2} . The curved surface area of the spherical zone—which excludes the top and bottom bases—is given by A = 2 π R h . {\displaystyle 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.

Region between parallel planes intersecting a sphere

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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 = π 6 h ( 3 a 2 + 3 b 2 + h 2 ) . {\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 b = 0 {\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

V = [ π a 2 ( h 2 ) ] + [ π b 2 ( h 2 ) ] + [ 4 3 π ( h 2 ) 3 ] {\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 h / 2 {\displaystyle h/2} ) and a sphere of radius h / 2 {\displaystyle h/2} .

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

A = 2 π R h . {\displaystyle 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

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• Spherical cap
• Spherical wedge
• Spherical sector

References

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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.

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

External links

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Wikimedia Commons has media related to Spherical segments.

• Weisstein, Eric W. "Spherical segment". MathWorld.
• Weisstein, Eric W. "Spherical zone". MathWorld.
• Summary of spherical formulas

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