Re: The perimeter and area of an ellipse
- From: "Dave L. Renfro" <renfr1dl@xxxxxxxxx>
- Date: Sat, 18 Oct 2008 08:04:19 EDT
Albert wrote:
derive that?Hi, the formula for an ellipse (perimeter and area) in my
notes are pi(a + b) and pi(ab) respectively. How'd you guys
Anon wrote:
Correct for area. The expression pi(a+b) might serve as an
acceptable approximation for the perimeter in some circumstances,
but it is not exact.
See, for example, (65) at <http://mathworld.wolfram.com/Ellipse.html>
for the exact formula for the perimeter (as well as some better
approximations). The same link also shows some methods to derive
the formula for the area.
Regarding the area of an ellipse, if the double integral change
of variable explanation at the mathworld webpage cited above
is beyond your current background knowledge, here's the essential
idea using high school ideas. (This is something I recently
wrote for another purpose, but since it could be of interest
here, I'm copying and pasting the relevant text here.)
Given the formula for the area of a circle, graph transformation
ideas from precalculus can be used to obtain a formula for the
area of an ellipse. The following outlines an argument that can
be made rigorous in a calculus course, but whose essential idea
can be conveyed using high school mathematics. An ellipse with
axes of lengths a and b can be placed in the coordinate plane
so as to be the graph of (x/a)^2 + (y/b)^2 = 1. If we apply the
transformation x --> ax' and y --> by', then the ellipse transforms
to the circle (x')^2 + (y')^2 = 1. We know the area of this circle
is pi. Now imagine the circle being covered by a very large number
of very tiny equal sized squares whose sides are parallel to the
coordinate axes. The sum of the areas of these squares will be
very close to the area of the circle. Moreover, by using sufficiently
small squares, we can approximate the area of the circle as closely
as we wish. Each of these tiny squares making up part of the area
of the circle corresponds, under the transformation above, to a
tiny rectangle covering part of the ellipse whose horizontal and
vertical sides are a and b times, respectively, the side length
of the square. Thus, the ellipse is covered by a large number of
tiny rectangles, each of which has an area that is ab times the
area of one of the squares. The sum of the areas of these rectangles
is ab times the sum of the areas of the squares (distributive law),
and this can be used to show that the area of the ellipse is ab
times the area of the circle, or pi*ab.
Dave L. Renfro
.
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- From: Anon
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