A spherical cap is the region of a sphere which lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is a called a hemisphere, and if the cap is cut by a second plane, the spherical frustum is called a spherical segment. However, Harris and Stocker (1998) use the term “spherical segment” as a synonym for what is here called a spherical cap and “zone” for spherical segment.

Let the sphere have radius , then the volume

of a spherical cap of height and base radius is given by the equation of a spherical

segment

(1) |

with ,

giving

(2) |

Using the Pythagorean theorem gives

(3) |

which can be solved for

as

(4) |

so the radius of the base circle is

(5) |

and plugging this in gives the equivalent formula

(6) |

In terms of the so-called contact angle (the angle between the normal to the sphere at the bottom of the cap and the base plane)

(7) |

(8) |

so

(9) |

The geometric centroid occurs at a distance

(10) |

above the center of the sphere (Harris and Stocker 1998, p. 107).

The cap height

at which the spherical cap has volume equal to half a

hemisphere is given by

(11) |

Consider a cylindrical box enclosing the cap so that the top of the box is tangent to the top of the sphere. Then the enclosing box has volume

(12) |

(13) |

(14) |

so the hollow volume between the cap and box is given by

(15) |

The surface area of the spherical cap is given by the same equation as for a general zone:

(16) |

(17) |