# volume of spherical cap and spherical sector

Theorem 1. The volume of a spherical cap^{} is $\pi {h}^{2}\left(r-\frac{h}{3}\right)$, when $h$ is its height and $r$ is the radius of the sphere.

Proof. The sphere may be formed by letting the circle ${(x-r)}^{2}+{y}^{2}={r}^{2}$, i.e.
$y=(\pm )\sqrt{rx-{x}^{2}}$, rotate about the $x$-axis. Let the spherical cap be the portion from the sphere on the left of the plane at $x=h$ perpendicular^{} to the $x$-axis.

Then the for the volume of solid of revolution yields the volume in question:

$$V=\pi {\int}_{0}^{h}{(\sqrt{rx-{x}^{2}})}^{2}\mathit{d}x=\pi {\int}_{0}^{h}(2rx-{x}^{2})\mathit{d}x=\pi \underset{x=0}{\overset{h}{/}}\left(r{x}^{2}-\frac{{x}^{3}}{3}\right)=\pi {h}^{2}\left(r-\frac{h}{3}\right).$$ |

Theorem 2. The volume of a spherical sector is $\frac{2}{3}\pi {r}^{2}h$, where $h$ is the height of the spherical cap of the spherical sector and $r$ is the radius of the sphere.

Proof. The volume $V$ of the spherical sector equals to the sum or difference of the spherical cap and the circular cone depending on whether $$ or $h>r$. If the radius of the base circle of the cone is $\varrho $, then

$$ |

But one can see that both expressions of $V$ are identical. Moreover, if $c$ is the great circle of the sphere having as a diameter^{} the line of the axis of the cone and if $P$ is the midpoint^{} of the base of the cone, then in both cases, the power of the point $P$ with respect to the circle $c$ is

$${\varrho}^{2}=(2r-h)h.$$ |

Substituting this to the expression of $V$ and simplifying give $V=\frac{2}{3}\pi {r}^{2}h$, Q.E.D.

Title | volume of spherical cap and spherical sector |

Canonical name | VolumeOfSphericalCapAndSphericalSector |

Date of creation | 2013-03-22 18:19:14 |

Last modified on | 2013-03-22 18:19:14 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 26B15 |

Classification | msc 53A05 |

Classification | msc 51M04 |

Synonym | volume of spherical cap |

Synonym | volume of spherical sector |

Related topic | SubstitutionNotation |

Related topic | GreatCircle |

Related topic | Diameter2 |

Related topic | PowerOfPoint |