Question

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon ( $7$ sides). The areas of the two regions were $A$ and $B,$ respectively. Each polygon had a side length of $2.$ Which of the following is true?

$A = \dfrac{25}{49}B$

$A = \dfrac{5}{7}B$

$A = B$

$A = \dfrac{7}{5}B$

$A = \dfrac{49}{25}B$

Solution:

For a regular polygon with side length $2,$ let $O$ be the center, $M$ the midpoint of a side, and $N$ an endpoint of that side. Then $\triangle OMN$ has a right angle at $M,$ with $MN = 1,$ $OM = r$ (inradius), and $ON = R$ (circumradius).

So $R^2 - r^2 = 1,$ and the area between the circles is $\pi(R^2 - r^2) = \pi$ for any number of sides. Hence $A = B.$

Thus, the correct answer is C.

2009 AMC 12A Problem 19

Solution:

Problem 19 in Other Years