HMMT 二月 2001 · 几何 · 第 4 题

英文解析

4. A circle has two parallel chords of length x that are x units apart. If the part of the circle included between the chords has area 2 + π , find x .
   Solution: Let C be the area of the circle, S be the area of the square two of whoseedges are the chords, and A be the area of the part of the circle included between the chords.
   √
   2 π
   2 2
   The radius of the circle is x , so C = x , and S = x . Then the area A is the area
   2 2
   of the square plus one half of the difference between the areas of the circle and square:
   √
   1+π
   C − S C + S 2 A
   A = + S = = x , so x = = 2 .22
   2 2 2 1+π 2