PlanetMath: easy calculation of the area of an ellipse

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easy calculation of the area of an ellipse

(Definition)

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 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 .

Now since the Jacobian of the transformation is constant, the change of variables in integral theorem allows us to say the area of the transformed set is times the area of the original set.

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

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Cross-references: Jacobian, axis, ellipse, circle, transformation, determinant, linear transformation, area, unit circle

This is version 4 of easy calculation of the area of an ellipse, born on 2006-03-06, modified 2007-04-22.
Object id is 7689, canonical name is EasyCalculationOfTheAreaOfAnEllipse.
Accessed 1838 times total.

Classification:

AMS MSC:

53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

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