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Show square inscribed in circle has maximum area

  1. Sep 6, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that the square, when inscribed in a circle, has the largest area of all the 4-sided polygons. Try to show that all sides of a quadrilateral of maximal area have to be of
    equal length.

    3. The attempt at a solution
    How do you start?
    I don't get how showing the sides are equal will help prove that the square has maximal area.
    Any hints?
  2. jcsd
  3. Sep 6, 2010 #2


    User Avatar

    Staff: Mentor

    Try to express surface of the inscribed 4-sided polygon as a function of the length of its sides. General case will be more difficult, but for a specific case of rectangle it should be relatively easy to express lengths as a function of one variable.

    Once you have that, find maximum.
  4. Sep 6, 2010 #3
    I don't quite understand.
    Say I have a square, let the side length = a.

    area = a^2.
    Then maximize by taking derivative?
  5. Sep 6, 2010 #4
    Real easy to prove with coordinate geometry, but you need to know the parametric equation of a circle and the area of a quadrilateral.

    Any point on a circle (with origin as centre and radius r) is (r cosx, r sinx)
    and area of a quadrilateral with vertices (x1, y1), (x2, y2), (x3, y3), (x4, y4),
    is #8 here http://www.mathisfunforum.com/viewtopic.php?id=3301

    take four points, put them in the formula, the result becomes obvious
  6. Sep 6, 2010 #5
    that gives me the area, but it doesnt prove that it is the maximum area for a 4 sided shape.
  7. Sep 6, 2010 #6
    it does! the result for area of a general quadrilateral in a circle comes to be (assuming radius r and angles A, B, C, D
    r2/2[sin(A-B) + sin(B-C) + sin(C-D) + sin(D-A)]
    the maximum value of which (2r2) is attained only when each of the angular differences is 90 degrees which results in a square

    did u try solving it? you need basic trignometric formulae
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