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Since the formula for the lateral area of a right cone is the same as the formula for the area of an ellipse, are there any deeper connections between these two geometric objects, other than the fact that an ellipse is a conic section?

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up vote 2 down vote accepted

Here is an intuitive explanation of why they should be equal:

Imagine taking a cone of radius $r$ and slant height $l$, removing the base of the cone, and putting it in a vise (the sides of the vise will be exactly $2r$ apart). Now squash the point of the cone down flat - the sides of the vise are preventing any expansion in one direction, so all of the extra "material" of the cone goes into extending it's footprint along the axis parallel to the sides of the vise. The result will be an ellipse with one radius equal to $r$ (this is perpendicular to the sides of the vise), and the other radius equal to the slant height (this is parallel to the sides of the vise).

So, the area of the "lateral area" part of a cone (i.e. the surface area, minus the base) is equal to the area of an ellipse with radii equal to $r$ and $l$.

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Is there an intuitive way to see why the right angle makes the squashing an isometry? – anon Sep 16 '11 at 10:23
I'm trying to think of one, but I think my brain is fried from lack of sleep. I just posted the first thing that came to mind. I would guess it's something like the vertical distance between points got converted to increasing their distance in the parallel-to-vise direction. – Zev Chonoles Sep 16 '11 at 10:26
Excellent. Thank you. I've up-voted and accepted your answer. – Mike Jones Sep 16 '11 at 19:24
BTW, I subsequently mentioned this to my students, so, you've had a direct and immediate impact on the mathematics education of some high school students. Is MSE great, or what? – Mike Jones Sep 20 '11 at 0:59
@Mike: It really is great! I'm glad to have helped. – Zev Chonoles Sep 20 '11 at 1:32

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