Two equal circle are drawn in square in such a way that a side of the square forms diameter of each circle. If the remaining area of the square is 42 cm², how much will the diameter of the circle measure ?

Let length of each side of square = 2π
According to the question,
$$\eqalign{ & \frac{{\pi {r^2}}}{2} + \frac{{\pi {r^2}}}{2} + 42 = {\text{Area of square}} \cr & \Rightarrow \pi {r^2} + 42 = 4{r^2} \cr & \Rightarrow 4{r^2} - \pi {r^2} = 42 \cr & \Rightarrow {r^2}\left( {4 - \frac{{22}}{7}} \right) = 42 \cr & \Rightarrow {r^2}\left( {\frac{{28 - 22}}{7}} \right) = 42 \cr & \Rightarrow \frac{{6{r^2}}}{7} = 42 \cr & \Rightarrow {r^2} = \frac{{42 \times 7}}{6} \cr & \Rightarrow {r^2} = 7 \times 7 \cr & \Rightarrow r = 7 \cr & \therefore 2r = 14\,cm \cr} $$