If a circle of radius r is inscribed in a n-sided regular polygon,and the perimeter is 2p,find the polygon's area.
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2 Answers
1If a circle is inscribed in a polygon ... then each angle between the diagonals meeting at the centre of the circle is 2∩/n radian...
The rad of circle given is r ...
Perimeter of circle is 2p = 2∩r .. r = p/∩ ...
On proceeding like this, ... the expr. comes out to be ...
length of half-diagonal of the polygon = h = r cos ∩/n
=p/∩ cos ∩/n
if n is large [ polygon ] .. therefore cos ∩/n is small
cos ∩/n can be approximated as ∩/n
Hence .. h = p/∩ * ∩/n = p/n
On further calculation ...
Hence area of the polygon = 2∩/n * p/∩ (p√(∩2-n2)n∩ )
= 2p2√(∩2-n2)∩n2
Hope there r no calculation mistakes ..... sorry if there r any !!!
11sorry...the perimeter of the polygon is 2p...you went on it right.
I've figured it out,thanks....