SOLUTION: There are two equilateral triangles with different areas, where O and O' are their circumcenters respectively. In the first, the distance OA is 9 cm, and in the second, the distanc

These equilateral triangles are similar (as any two equilateral triangles,
by three angles). 

Their corresponding similar elements are OC and O'C'.


In the second triangle, we are given  O'C' =  cm.


In the first triangle, we are given OA = 9 cm.



      But triangle OAC is a right-angled 90°-60°-30°-triangle;

      therefore, the hypotenuse OC is twice the leg OA,

      so, the hypotenuse OC is 2*9 = 18 cm.



Thus, the ratio of corresponding linear dimensions in triangles is

    OC        18            9         9*sqrt(2)
   ----- = ----------- = -------- = ----------.
    O'C'    4*sqrt(2)     2*sqrt(2)      4


The ratio of the areas of the second triangle to the first triangle is the square
of the ratio of their corresponding linear elements, i.e.

     =  =  = 10.125.    ANSWER