Question

One circle is inscribed in, and a second circle is circumscribed about, a regular polygon of 2012 sides. If the polygon’s perimeter is 2012, what is the area of the region between the two circles?

Answer 1

each side has length 1


x is the radius of the bigger circle,
y is the radius of the smaller circle,
\(\theta\) is the angle between x and y.

Answer 2

\[\large \theta = \frac 12 \times {2\pi \over2012 }= {\pi \over2012}\]

Answer 3

\[\tan \theta = \frac {.5} y\]\[\implies y = {0.5\over \tan \left( \pi/2012 \right)}\]

Answer 4

\[\sin \theta= {0.5 \over x}\]

Answer 5

area between circles \[=\pi x^2- \pi y^2\]