# easy calculation of the area of an ellipse

Consider the unit circle $\left\{\right(x,y)\in {\mathbb{R}}^{2}:{x}^{2}+{y}^{2}\le 1\}$. It’s a well known fact that the area of this set is $\pi $.

Now consider the following linear transformation $(x,y)\to (u,v)=(ax,by)$.

The determinant^{} of the transformation is $ab$ and the transformed circle is:

$\left\{\right(u,v)\in {\mathbb{R}}^{2}:{\left(\frac{u}{a}\right)}^{2}+{\left(\frac{v}{b}\right)}^{2}\le 1\}$ an ellipse of axis $(a,b)$.

Now since the Jacobian^{} of the transformation is constant, the change of variables in integral theorem (http://planetmath.org/ChangeOfVariablesInIntegralOnMathbbRn) allows us to say the area of the transformed set is $ab$
times the area of the original set.

Thus, the area of an ellipse is $\pi ab$.

Title | easy calculation of the area of an ellipse |
---|---|

Canonical name | EasyCalculationOfTheAreaOfAnEllipse |

Date of creation | 2013-03-22 15:44:18 |

Last modified on | 2013-03-22 15:44:18 |

Owner | cvalente (11260) |

Last modified by | cvalente (11260) |

Numerical id | 7 |

Author | cvalente (11260) |

Entry type | Definition |

Classification | msc 53A04 |