All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
Two circles (Posted on 2004-10-21) Difficulty: 3 of 5
In a 8½x11 sheet of paper I drew two equal non-overlapping circles -- both completely inside the paper, of course.

What's the largest portion of the paper I could cover with the circles?

What would be the answer if I drew THREE equal circles?

No Solution Yet Submitted by Federico Kereki    
Rating: 3.2500 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Two circles case | Comment 1 of 11

Two circle case:

Note that we can assume the two circles to be tangent, since if they were not, we could slide them towards their center of gravity while keeping them within the paper.  We can then slide one of the circles, while maintaining the point of tangency between the two circles, so that it touches two adjacent edges of the paper.  This still keeps them within the paper.  Finally, we can revolve the other circle around the fixed circle so that it too is tangent to two adjacent edges of the paper, dilating the two circles if necessary.  The end result leads to three possible optimal configurations:  two where the two circles share a common tangent edge, and a third where they don't.  Computing these configurations yields that the third is the optimal one, with a radius of (39-2sqrt(187))/4 = 2.9126...

(the computation is done easily by using coordinates, using the fact that the two circles pass through the center of the paper by symmetry; details have been omitted)

  Posted by David Shin on 2004-10-21 17:08:16
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2019 by Animus Pactum Consulting. All rights reserved. Privacy Information