Re: polygons circumscribing a circle
- From: se16@xxxxxxxxxxxxxx
- Date: Wed, 18 Jul 2007 09:36:03 -0700
On 16 Jul, 19:35, Musing <generaliz...@xxxxxxxxxxx> wrote:
How does one show that the perimeter of a regular polygon
circumscribing a circle is greater than the circumference (of the
circle)? Thanks.
Take n as the number of sides of the polygon and r as the circle
radius. Join the centre to the vertices and edge mid-points of the
polygon. This gives you 2n congruent right angled triangles and 2n
congruent circle sectors. Take t as the angle of each of these at the
centre. Then the circumference of the polygon is 2nr*tan(t) while the
cicumference of the circle is 2nr*t. Since t is positive but less
than a right angle, we have tan(t)>t and hence the requested result.
.
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