Four circles having equal radii are drawn with center at the four corners of a square. Each circle touches the other two adjacent circle. If remaining area of the square is 168 cm, what is the size of the radius of the radius of the circle? (in centimeters)

Diameter of circle = side of square = $$d$$ cm

Area of square = $$d^2$$ sq. cm

Area of 1 quadrant = $$\frac{1}{4} \times \pi \times (\frac{d}{2})^2$$

= $$\pi \times \frac{d^2}{16}$$

=> Area of 4 quadrants = $$4 \times \pi \times \frac{d^2}{16} = \frac{\pi d^2}{4}$$ sq. cm

Area of shaded region = 168

=> $$d^2 - \frac{\pi d^2}{4} = 168$$

=> $$\frac{1}{4} [d^2 (4 - \pi)] = 168$$

=> $$d^2 = \frac{168 \times 4}{4 - \pi} = \frac{168 \times 4}{4 - \frac{22}{7}}$$

=> $$d^2 = \frac{168 \times 4 \times 7}{28 - 22} = \frac{168}{6} \times 28$$

=> $$d = \sqrt{28 \times 28} = 28$$ cm

$$\therefore$$ Radius = $$\frac{28}{2} = 14$$ cm