Finding the area

The shaded region consists of one square and four segments of a circle. To get the area of the segment, compute first the area of the sector and area of the triangle. Then to get the length of side $x$ , use cosine law. Note that the central angle of the sector is $30^\circ$ .

The area of the sector is $\dfrac{30}{360}\pi(10^2) = \dfrac{25}{3}\pi$ .

The area of the triangle is $\dfrac{1}{2}(10^2)(sin~30)=25$

The area of one segment is $\dfrac{25}{3}\pi-25$ .

By cosine law, the area of the square is

$x^2=10^2+10^2-2(10)(10)(cos~30)=26.79491924$

The area of the shaded region is $26.79491924+4(\dfrac{25}{3}\pi-25)=$ $\boxed{\color{#3D99F6}31.515}$