MathDB
Sector Area Ratio

Source: AMC 12 2008B Problem 4

February 29, 2008
geometryratioAMC

Problem Statement

On circle O O, points C C and D D are on the same side of diameter AB \overline{AB}, \angle AOC \equal{} 30^\circ, and \angle DOB \equal{} 45^\circ. What is the ratio of the area of the smaller sector COD COD to the area of the circle? [asy]unitsize(6mm); defaultpen(linewidth(0.7)+fontsize(8pt));
pair C = 3*dir (30); pair D = 3*dir (135); pair A = 3*dir (0); pair B = 3*dir(180); pair O = (0,0); draw (Circle ((0, 0), 3)); label ("CC", C, NE); label ("DD", D, NW); label ("BB", B, W); label ("AA", A, E); label ("OO", O, S); label ("4545^\circ", (-0.3,0.1), WNW); label ("3030^\circ", (0.5,0.1), ENE); draw (A--B); draw (O--D); draw (O--C);[/asy]<spanclass=latexbold>(A)</span> 29<spanclass=latexbold>(B)</span> 14<spanclass=latexbold>(C)</span> 518<spanclass=latexbold>(D)</span> 724<spanclass=latexbold>(E)</span> 310 <span class='latex-bold'>(A)</span>\ \frac {2}{9} \qquad <span class='latex-bold'>(B)</span>\ \frac {1}{4} \qquad <span class='latex-bold'>(C)</span>\ \frac {5}{18} \qquad <span class='latex-bold'>(D)</span>\ \frac {7}{24} \qquad <span class='latex-bold'>(E)</span>\ \frac {3}{10}