Little Area

A right triangle can be constructed where the hypotenuse is the segment joining the center of the little red circle and the center of one of the larger circles, as pictured below.

If the little red circle’s radius is r, then the hypotenuse is the sum of the two radii or $4 + r$ , the vertical leg is the difference of the two radii or $4 - r$ , and the horizontal leg is half the length of the square or $2$ .

Using the Pythagorean Theorem gives $(4 - r)^2 + 2^2 = (4 + r)^2$ , and solving this gives $r = \frac{1}{4}$ . The area of the little red circle must therefore be $A = \pi r^{2}\ = \pi (\frac{1}{4})^{2}\ = \frac{\pi}{16}$ .