A spherical segment is the solid defined by cutting a sphere 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 a zone. However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for spherical cap and "zone" for what is here called a spherical segment.
Call the radius of the sphere and the height of the segment (the distance from the plane to the top of sphere) . Let the radii of the lower and upper bases be denoted and , respectively. Call the distance from the center to the start of the segment , and the height from the bottom to the top of the segment . Call the radius parallel to the segment , and the height above the center . Then ,
(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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Relationships among the various quantities include
(7)
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(8)
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(9)
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(10)
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(11)
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Plugging in gives
(12)
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(13)
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(14)
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The surface area of the zone (which excludes the top and bottom bases) is given by
(15)
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