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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)

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Solution Part Two Solution | Comment 3 of 10 |

Let t be the measure of the angle between the line
joining the centers of the semicircles and the
straight edge of one of the semicircles. t must lie
in the  interval [30,90]. Clearly, for the smallest
rectangle, the semicircles should touch.
Therefore,

  A = 4 sin(t)[1 + cos(t)]             (1)

At t=30, A = 2 + sqrt(3) ~= 3.7320508

At t=90, A = 4

Setting dA/dt = 0 gives

  2 cos(t)^2 + cos(t) - 1 = 0          (2)

Therefore,

  cos(t) = 1/2

  t = 60 degrees

Plugging these values into (1) gives

  A = 3 sqrt(3) ~= 5.1961524

Clearly, the smallest rectangle occurs
at t=30 and the ratio is

  A/4 = 0.9330127

Note: Each, of the opposite sides of the enclosing
      rectangle, contains the straightedge of a
      semicircle and is tangent to the other
      semicircle.

 


  Posted by Bractals on 2007-08-22 12:49:09
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