Simple area as it looks!

It can be easily seen from the figure above that the area of the shaded region is half of the square. Since the area of the square is 1 that of the shaded area is $\boxed {\dfrac 12}$ .

Draw the diagonals and you will see that the oval is half the diagonal so this means it will fit on length. Split it in two parts by the diagonal and rotate the parts until they fit. You will now see two triangles one withe and one red like Syed Hamza Khalid solution which are half the square or $\frac{1}{2}$ .