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 
,
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(1)
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(2)
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 |  | ![pi[R^2y-1/3y^3]_d^(d+h)](/images/equations/SphericalSegment/Inline18.svg) |
(3)
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 |  | ![pi{R^2h-1/3[(d+h)^3-d^3]}](/images/equations/SphericalSegment/Inline21.svg) |
(4)
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 |  | ![pi[R^2h-1/3(d^3+3d^2h+3h^2d+h^3-d^3)]](/images/equations/SphericalSegment/Inline24.svg) |
(5)
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(6)
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Relationships among the various quantities include
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(7)
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(8)
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(9)
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(10)
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 |  | ![sqrt(([(a-b)^2+h^2][(a+b)^2+h^2])/(4h^2)).](/images/equations/SphericalSegment/Inline42.svg) |
(11)
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Plugging in gives
 |  | ![pih[1/2(a^2+b^2+h^2)-1/3h^2]](/images/equations/SphericalSegment/Inline45.svg) |
(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
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(15)
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