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 $a b$ 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 $a b$ 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