User:Lenparhs/sandbox

Introduction
Cornford's Carrot Corollary is small a theorem that can be used to calculate the amount of wastage that is created when peeling a carrot. The corollary was conceived by Gavin Cornford when he was peeling carrots. He realised that the difference between the area of a circle and an n-sided regular polygon created by peeling a carrot with n strokes could be calculated using the some basic trigonometry. The corollary yields some quite interesting and counter-intuitive results.

The Formula
The formula is derived by calculating the area of the segment that is peeled away from the carrot. This area is given by:

$$ \frac{\theta}{360}\pi r^2 - \frac{1}{2} r^2 \sin \theta $$ where $$r$$ is the radius, and $$ \theta $$ is the angle of the sector.

This can be factorised to yield:

$$ r^2 \left( \frac{\theta \pi}{360} - \frac{1}{2} \sin \theta \right) $$