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perimeter, approaching but never reaching the area of a circle with the same circumference (last entry could be a circle). The existing chart keeps the side lengths at 1 unit, which gives astronomical areas for higher-number n-gons but doesn't make it easy to directly compare the areas of the various polygons. Thanks.
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I have also removed the formula -7 cos(2 Pi/7) / ( 4 sin(2 Pi/7) - 4 ) for the heptagon. It doesn't bring anything new and is more tedious to calculate than the general cot-formula. Furthermore that expression should haven been simplified to 7/4 cos(2 Pi/7) / ( 1 - sin(2 Pi/7) ) and then to 7/4 cot(
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The paragraph after the table says: "The amounts that the areas are less than those of circles with the same perimeter, are (rounded) equal to . . . ." As a non-mathematician Math
Olympiad coach, I'd love to see a table or graph here showing the increasing area of regular polygons of equal
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I added \tfrac{p^2}{4n} \cot{\tfrac{\pi}{n}} as it was the form I was looking for when I loaded the page (and none of the others are in terms of just p,n). Is general policy to avoid any redundant formulae? If so I'm not sure why nsa/2 is there as it's also clearly pa/2.
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I recently added a theorem proved in 2004 by Hwa Young Lee about origami constructible polygons (she also proved many other things in the paper; it's worth a read if you haven't). I'm not sure I formatted the citation correctly. Help would be appreciated!
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Also, shouldn't the "exact area" for the square be the formula s^2 and the "approximate area" be "1"? Seems like the "1" is in the wrong column, even though it is exact, not approximate. Thanks.
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gives the simplest form using the perimeter. Your addition was a second application of this and I did not see the need for it as it did not add anything novel to the string of formulae.--
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There is no reason why we should fear the cot-function and replace it with cos/sin. The latter formula does not belong in the approximated values column, not even with parenteses.
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What exactly is the problem? In a regular polygon an interior angle cannot exceed 180°, since that would form a straight line (without a vertex). This is not rocket science.
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simplify, but we just don't know. One of the non-empty cells had the marker as well. I.m.o. an empty cell is really ok. We don't have to put reasons for being empty in there.
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There are literally thousands of ways these formulas can be written, so one needs to make some choices as to which ones should be given. The formula
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In the section on diagonals, I suggest that it say that the number of different diagonal lengths of a regular convex n-gon is floor(n/2)-1.
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you should focus on finding relations which are commonly discussed / considered important in existing literature, not just which are true. –
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is present because it is the clearest link to the area of a triangle form and so, the most basic version. Using
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The general formula should come first. Ssgxnh moved it back to the bottom. I'd like to insist having first.
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Knowledge. If you would like to participate, please visit the project page, where you can join
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Ssgxnh, please propose and discuss on talk page before you make further changes? - Thanks -
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Twice I undid this unsourced, tirivial addition that an internal angle is smaller than pi:
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Some of the exact values are either unknown, or perhaps not expressable in radical form. -
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This does not seem worth mentioning to me, beyond the existing discussion in
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Is the table incomplete? Some of the exact areas in the table aren't listed.
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RIght, and 1+1=2. Not rocket science either. Notability is key here. -
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