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## Discussion of Problem 1020. Rope

mathematical proof
Posted by emilchess 6 Nov 2012 02:31
Draw a polygon. the sum of its angles is pi*(n-2). The sum of pieces of circles is R*(n*pi-(n-2)*pi)=2*R*pi (you can see it easily if u draw)
Re: mathematical proof
Posted by Ouch 7 Nov 2013 14:20
Thanks, so nice.
Re: mathematical proof
Posted by prostack 2 Feb 2014 08:46
yep, its simple
Re: mathematical proof
Posted by lakerka 22 Mar 2015 17:54
Beautiful. Solved it by calculating angles between vectors of consecutive nails.
Re: mathematical proof
Posted by Abdul Matin 6 Oct 2018 22:57
How is the sum of its angles pi*(n-2)?
Re: mathematical proof
Posted by crvdgc 10 Mar 2019 18:29
OP means the sum of angles in a n-polygon is pi*(n-2). This can be attained inductively:
1. The sum of angles in a triangle is pi.
2. for any n-polygon, it can be divided into a (n-1)-polygon and a triangle.
Re: mathematical proof
Posted by _confused 3 Dec 2020 21:53
thanks !