Quandaries
and Queries 

Hi my name is daniel, I am 14 and i have been given a piece of maths coursework whereby a farmer has to fence off a piece of land as large as possible using 1000m of fence. I already know that the formula for working out the area of any shape of a 1000m perimeter = 500^{2}/ n*tan*(180/n), however, after some research I have found out that as the number of sides (n), tends to infinity, the n*tan*(180/n) tends to pi. Why is this? If you could email me the answer it would be great. 

Hi Daniel, The key to this is the fact that,
I am not going to prove this for you, there is a proof in the Math Archives at the University of Tennessee in Knoxville Tennessee. 180 degrees is radians and hence ^{180}/_{n} degrees is ^{}/_{n} radians. Hence, rewriting your expression with the angle expressed in radians rather than degrees gives
Now let t = ^{}/_{n} and you expression can again be rewritten, this time as
But n = ^{}/_{t} and hence
Now, as n tends to infinity, t tends to zero and hence ^{sin(t)}/_{t} tends to 1 and cos(t) tends to 1. Thus your expression tends to
Penny 

