All curves

The angle is a bit of a red herring. Changing it will not alter the area of the shape (unless it is too small.) Change it to 180 degrees and color as so:

We see the light blue segments can be moved to the light green to form four green squares each of area $\sqrt{2}^{2}=2$ . The red segment on one end can move to the other side to complete a single circle of unit radius.

Total area: $8+\pi \approx \boxed{11.14159265}$

The shape can be subdivided into $3$ sets of congruent shapes:

There are $16$ pink shapes that are $\frac{1}{8}$ unit circles for an area of $16 \cdot \frac{1}{8} \pi \cdot 1^2 = 2 \pi$ .

There are $4$ blue shapes that are the difference between a unit square and a $\frac{1}{4}$ unit circle for an area of $4(1 - \frac{1}{4} \pi \cdot 1^2) = 4 - \pi$ .

There are $4$ yellow shapes that are unit squares for an area of $4 \cdot 1^2 = 4$ .

This gives a total area of $2 \pi + 4 - \pi + 4 = 8 + \pi \approx \boxed{11.1415925}$ .