Snake eyes

The diagram below shows the eye and also the two full circles whose arcs form the slit, and some lengths.

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Notice that all of the angles in the triangles formed are $90^\text{o}$ or $45^\text{o}$. By applying Pythagoras' theorem to the larger triangle (or any of the right-angled triangles), $x^2+x^2=2^2\Rightarrow 2x^2=4\Rightarrow x^2=2$, so $x=\sqrt2$.

In the diagram below, the grey area and the green area together form a sector of the left hand circle. The angle at the middle of the sector is $90^\text{o}$, so the sector is a quarter of the whole circle.

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The area of the green triangle is $\frac{1}{2}\times 2\times1=1$ cm$^2.$

The radius of the larger circle is $x=\sqrt2$ cm, so the area of the larger circle is $\pi\times\sqrt{2}^2=2\pi$ cm$^2.$ So the area of the sector is $\frac{1}{4}\times2\pi=\frac{\pi}{2}$ cm$^2.$

So the area of the grey segment is $\frac{\pi}{2}-1$ cm$^2.$

The yellow area is just the area of the eye without twice the grey area. The area of the eye is $\pi1^2=\pi$ cm$^2$. So the yellow area must be $\pi-2(\frac{\pi}{2}-1)=\pi-\frac{2\pi}{2}+2=2$ cm$^2.$