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Slice a Circle to fit a Rectangle (Posted on 2007-08-22) Difficulty: 4 of 5
The goal of this problem is to slice a unit circle into two pieces which can be fit into a rectangle of minimal area.
Consider these three methods:

One: Slice the circle across its diameter and slide the pieces along each other a little.
What distance between the radii of the two semicircles gives the smallest rectangle? This rectangle has a smaller area than square that circumscribes the original circle. What is the ratio of rectangle to square?

Two: Slice the circle along a diameter and place these into a rectangle so the straight edges of each semicircle are along opposite edges of the rectangle.
What is the ratio of rectangle to square this time?

Three: Slice a segment off of the circle and place this segment to the side. The large piece's straight edge is along one side of the rectangle. The straight edge of the segment is tangent to the large piece but not necessarily parallel to a side of the rectangle.
What chord length minimizes the rectangle? What is the ratio?

Note: Part Three is may be particularly difficult to find an exact solution for. If you have a method of finding a good approximation feel free to share your results.

No Solution Yet Submitted by Jer    
Rating: 4.1667 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Part Three Results | Comment 9 of 10 |

I get the following from a Perl program.
The values agree with what I get from Sketchpad:

Length of chord

= 1.61465519                                                                                                          

Ratio

= 0.95332890      

The endpoints of the segment lie on the opposite edges
of the rectangle.

The edge of the rectangle opposite the edge containing
the large piece's straight edge is tangent to the large
piece.

The central angle between the point of tangency and the
endpoint of the large piece's straight edge is

  46.16873531 degrees.

It looks like Part Two wins so far.
    


  Posted by Bractals on 2007-08-24 15:59:53
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