2002 AMC 10A Problems/Problem 7
Problem
A arc of circle A is equal in length to a arc of circle B. What is the ratio of circle A's area and circle B's area?
Solution
Let and be the radii of circles A and B, respectively.
It is well known that in a circle with radius r, a subtended arc opposite an angle of degrees has length $\frac{\theta}{360}\cdot{2\pi{r}$ (Error compiling LaTeX. Unknown error_msg).
Using that here, the arc of circle A has length . The arc of circle B has length $\frac{30}{360}\cdot{2\pi{r_2}=\frac{r_2\pi}{6}$ (Error compiling LaTeX. Unknown error_msg). We know that they are equal, so , so we multiply through and simplify to get . As all circles are similar to one another, the ratio of the areas is just the square of the ratios of the radii, so our answer is .