easy calculation of the area of an ellipse

Consider the unit circle {(x,y)2:x2+y21}. It’s a well known fact that the area of this set is π.

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

The determinantMathworldPlanetmath of the transformation is ab and the transformed circle is:

{(u,v)2:(ua)2+(vb)21} an ellipse of axis (a,b).

Now since the JacobianMathworldPlanetmath 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 πab.

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