# area of spherical calotte by means of chord

Let the arc $PR$ of a circle with radius $r$ rotate about the diameter^{} $PQ$. The surface of revolution^{} is a spherical calotte with the height $h$. If the of the chord $PR$ is $k$, we obtain from the right triangle^{} $PQR$ the proportion equation

$$\frac{h}{k}=\frac{k}{2r},$$ |

i.e. the chord $k$ is the central proportional of the height and the diameter. Accordingly, we can substitute $2rh={k}^{2}$ to the expression

$$A=2\pi rh$$ |

of the area of the spherical calotte derived in the parent entry (http://planetmath.org/AreaOfSphericalZone). Thus we have an alternative

$A=\pi {k}^{2}$ | (1) |

for finding the area of a spherical calotte.

## References

- 1 K. Väisälä: Geometria. Kymmenennen painoksen muuttamaton lisäpainos. Werner Söderström Osakeyhtiö, Porvoo & Helsinki (1971).

Title | area of spherical calotte by means of chord |
---|---|

Canonical name | AreaOfSphericalCalotteByMeansOfChord |

Date of creation | 2013-03-22 18:19:20 |

Last modified on | 2013-03-22 18:19:20 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Derivation |

Classification | msc 51M04 |

Synonym | alternative way to find area of spherical calotte |

Related topic | ThalesTheorem |

Related topic | SimilarityOfTriangles |