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Topic: Area of Sphere = 4 * (Area of Circle)
Replies: 7   Last Post: Dec 15, 2003 6:16 PM

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Daniel A. Asimov

Posts: 101
Registered: 12/3/04
Area of Sphere = 4 * (Area of Circle)
Posted: Jun 25, 1996 12:45 PM
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On the Usenet newsgroup "sci.math", someone recently inquired why the area of a
sphere is equal to 4 times the area of a disk of the same radius.

Using calculus, it is of course straightforward to prove the area formulas
area(disk) = pi r^2 and area(sphere) = 4 pi r^2, whence the result follows.

But does anyone know a more direct way to see that

area(sphere) / area(disk) = 4 ???

A decomposition of a disk into a finite number of "nice" pieces that can
be reassembled into a quarter-sphere seems out of the question, since the
disk has Gaussian curvature = 0, while the sphere's curvature is 1/(r^2).

HOWEVER, maybe there is a convincing decomposition of the disk into an
infinite number of "nice" pieces, each of which has 0 area, and which can
be reassembled into a quarter sphere.

This method is probably fraught with pitfalls, but perhaps there is a
valid approach like this which could somehow be made rigorous.

(For simplicity, let's assume from now on that r = 1.)

--Dan Asimov






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