The area of an ellipse is: [math]A= \pi ab [/math] where [math]a[/math], and [math]b[/math] are the major and minor axes of the ellipse. Isn't that just wonderful?!?! It's not at all obvious, but at the same time it is just what you would hope it would be. It's definitely near the top of my list of favorite formulas. [br][br]This applet explains a way to derive this formula and also gives a visual representation to help make sense of that derivation. The figure shows an ellipse inside a circle such that they share a center and the radius of the circle, [math]R[/math] is equal to the major axis, [math]a[/math] of the ellipse.[br][br]What makes this even more amazing is that the circumference of the ellipse is as complicated as the area is simple. It can only be expressed using integrals!!