Skip to main content

2009 AMC 12A Problem 19

Problem 19 of 25HarderGeometry

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 (77 sides). The areas of the two regions were AA and B,B, respectively. Each polygon had a side length of 2.2. Which of the following is true?

Answer choices

Show solution

Solution

For a regular polygon with side length 2,2, let OO be the center, MM the midpoint of a side, and NN an endpoint of that side. Then △OMN\triangle OMN has a right angle at M,M, with MN=1,MN = 1, OM=rOM = r (inradius), and ON=RON = R (circumradius). So R2−r2=1,R^2 - r^2 = 1, and the area between the circles is π(R2−r2)=π\pi(R^2 - r^2) = \pi for any number of sides. Hence A=B.A = B. Thus, the correct answer is C.
AoPS wiki

Tagged: regular polygon · Pythagorean Theorem · circle area

More practice