Prove that among all polygons, of fixed side, a circle has the maximum area!
62Hint Use the trick that was used to prove that among all quadrilaterals, square has maximum area!
62Nothing to know about geometry here... or should i say hardly anything
whatever is needed is in the Trigonometry syllabus!
39If we then go to an n-sided figure with perimeter P, the area of one
of the triangle formed by joining the midpoint of the figure
to the vertices is:
(1/2)r x cos(pi/n) x P/n
where r = distance from centre point to a vertex),
and with n such
triangles the area is (1/2)r x P x cos(pi/n).
This will tend to its greatest value as cos(pi/n) -> 1 which requires
that n -> infinity.
So we in the end get a polygon of an infinite number
of sides, which is a circle.
39ok, we have a theorem which states of all polygons, regular polygon has maximum area.
11shayad sir aapne isse dekha nahin
but bharghav has given this link in chatbox
http://www.mathlinks.ro/viewtopic.php?t=264064
62What am i supposed to do with this?
@b555.. You have proved only that among regular polygons, circle has highest area..
You have proved nothing for "All Polygons"
39sirry that link was only inspirit of a good problem, nothing to do with question here
ok i am trying for all polygons
39WHAT I DID WAS MAY BE A BIT LONG BUUT CORRECT METHOD.
AMONG ALL POLYGONS , REGULAR POLYGONS HAVE HIGHEST AREA AND AMONG ALL REGULAR POLYGONS CIRCLE HAS MAXIMUM AREA, HENCE
AMONG ALL POLYGONS, CIRCLE HAS MAXIMUM AREA.
9take a loop of given lenght and extend it outwards to get max area itll be a circle