The figure shows a square with four circles of equal area, each touching two sides of the square and two other circles. What is the ratio between the areas of the black region and the gray region? Note that each white circle is inscribed within a quarter of the larger square, as divided by the red lines. Then we can note that each quarter has one black part and three gray parts, and the black part is equal to the size of one gray part, and therefore the ratio of areas is 1 to 3 feet.

square

circles

area ratio

black region

gray region