EasyCalculationOfTheAreaOfAnEllipse 2006-03-06 16:31:06 2007-04-22 11:30:26 Definition ellipse % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \PMlinkname{change of variables in integral theorem}{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 $\pi a b$.