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 $k$.
Maximum area with a given perimeter
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geometry
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1Is 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 – 2011-08-27
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1How 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. – 2011-08-27
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2*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. – 2011-08-27