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) |
 |  |  | (2) |
 |  | ![pi[R^2y-1/3y^3]_d^(d+h)](https://mathworld.wolfram.com/images/equations/SphericalSegment/Inline18.svg) | (3) |
 |  | ![pi{R^2h-1/3[(d+h)^3-d^3]}](https://mathworld.wolfram.com/images/equations/SphericalSegment/Inline21.svg) | (4) |
 |  | ![pi[R^2h-1/3(d^3+3d^2h+3h^2d+h^3-d^3)]](https://mathworld.wolfram.com/images/equations/SphericalSegment/Inline24.svg) | (5) |
 |  |  | (6) |
Relationships among the various quantities include
 |  |  | (7) |
 |  |  | (8) |
 |  |  | (9) |
 |  |  | (10) |
 |  | ![sqrt(([(a-b)^2+h^2][(a+b)^2+h^2])/(4h^2)).](https://mathworld.wolfram.com/images/equations/SphericalSegment/Inline42.svg) | (11) |
Plugging in gives
 |  | ![pih[1/2(a^2+b^2+h^2)-1/3h^2]](https://mathworld.wolfram.com/images/equations/SphericalSegment/Inline45.svg) | (12) |
 |  |  | (13) |
 |  |  | (14) |
The surface area of the zone (which excludes the top and bottom bases) is given by
 | (15) |
