jmc

Question

Two concentric circles are centered at point P. The sides of a 45 degree angle at P form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 36 degree angle at P. What is the ratio of the area of the smaller circle to the area of the larger circle? Express your answer as a common fraction.

Accepted Answer

Let C_1 and C_2 be the circumferences of the smaller and larger circle, respectively. The length of the 45^ arc on the smaller circle is (45^360^)C_1, and the length of the 36^ arc on the larger circle is (36^360^)C_2. Setting these two lengths equal we find C_1C_2=3645=45. The ratio of the areas of the two circles is the square of the ratio of their circumferences: r_1^2 r_2^2=(r_1r_2)^2=(2 r_12 r_2)^2=(C_1C_2)^2=(45)^2=1625.