What is the area of the marked region?



The four quarter of circles cover the square.
They overlap in four copies of the marked region.
Each quarter circle has area \(\frac{\pi}{8}\) and the square has area \(1\).
So the marked region has area \[\frac{1}{4}\cdot \big(4 \cdot \frac{\pi}{8}-1\big)=\dfrac{\pi}{8}-\frac{1}{4}\,.\]