Intresting question

 Four circles of equal size are inscribed in a square as shown in the diagram. Inside of the four circles is a smaller square tangent to each of the four circles.

          If the large square has a side length equal to 4, what is the area of the small square?


solution:-The side of the large square has a length equal to 4 times the radius of a circle, which can be seen by drawing a straight line segment that connects the centers of two adjacent circles. Thus each circle has a radius equal to 1.
 

The trick to this problem is creating a right triangle that relates the relevant lengths.
Connect the centers of 3 adjacent circles. The new shape is an isosceles right triangle, where each leg equals 2. This means the hypotenuse has a length 2√2.
The hypotenuse is also equal to the length of two radius lengths plus the side of the small square, x.




Thus we have:
x + 2 = 2√2
x = 2√2 – 2
The area is then the square of this side.
Area = x² = (2√2 – 2)² = 12 – 8√2 ≈ 0.686


By-vikash Rahii