# easy calculation of the area of an ellipse

Consider the unit circle $\left\{\right(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}\leq 1\}$. 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:

$\left\{\right(u,v)\in\mathbb{R}^{2}:\left(\frac{u}{a}\right)^{2}+\left(\frac{v% }{b}\right)^{2}\leq 1\}$ 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 EasyCalculationOfTheAreaOfAnEllipse 2013-03-22 15:44:18 2013-03-22 15:44:18 cvalente (11260) cvalente (11260) 7 cvalente (11260) Definition msc 53A04