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An arbitrary pentagon (which is convex) is given to you, and it has a perimeter of $k$. Determine the i) maximum area, rigorously, in terms of $k$. ii) maximum INTEGRAL area, in terms of $k$.

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Is it that u want to know when the area can be maximum? if so its in the case of a regular pentagon and is given by {n(l^2)}/4tan(180/n) where n is number of sides and t is length of each side so nt=k –  Bhargav Aug 27 '11 at 17:40
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How rigorously? If we assume that there is a pentagon of perimeter $k$ which has maximum area, showing that this optimal pentagon is regular is not difficult. However, although the existence of an optimal pentagon is intuitively reasonable, proving it rigorously takes quite a bit of effort. –  André Nicolas Aug 27 '11 at 19:52
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Among all $n$-gons inscribed in a circle, which one has largest area and which one has largest perimeter? The regular polygon is the expected answer, and this turns out to be the case. Maxima and minima without calculus by Ivan Morton Niven Volume 6, 6.1 Introduction. –  Américo Tavares Aug 27 '11 at 21:52

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