The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Math Topics »

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Daniel A. Asimov

Posts: 101
Registered: 12/3/04
Area of Sphere = 4 * (Area of Circle)
Posted: Jun 25, 1996 12:45 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.