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Re: [Phys-l] elliptical thinking

On Jul 21, 2008, at 2:46 PM, John Denker wrote:

Hi Folks --

Quick question:

What's the formula for the area of an ellipse?

More interesting question: How do you know?
-- Do you remember the formula from high-school geometry?
-- Did you look it up just now?
-- Or do you have some other way of knowing?

I recently wrote up my notes on one way of figuring it out:

I like this way because
a) It illustrates a particular type of scaling argument that is
super-easy but often under-emphasized.
b) It serves as a good excuse for a riff on "figuring things out"
in general, and its relationship to memorization.

This also makes contact with our recent discussions of "new math"
and its relation to "old math":

This issue is commonly referred to as the “Math Wars” but I don’t
like to use that term. The warlike aspects are a discredit to
everyone involved. The sensible approach is to use smart, efficient
algorithms *and* to understand the principles involved.

1) I am replying to John's message without looking at what he wrote. I do remember that S= pi*a*b. But I do not remember when and where I learned this.
2) Here is an interesting observation. Suppose I want to determine the lower limit, to be labeled as S'. I sketch four identical right triangles, inside of the ellipse. Thus S'=4*(a*b/2)=2*a*b.
3) Then I do the same for circle. This time S'=4*(R*R/2=2*R^2.
4) It is interesting that in each case S' becomes S if 2 is replaced by pi.

Yes, I know, this is only an observation. John D. is asking for a derivation.

Ludwik Kowalski, a retired physics teacher
5 Horizon Road, Apt. 2702, Fort Lee, NJ, 07024, USA
Also an amateur journalist at