Area

Draw segment EC and denote the area of triangle FCE by $x$ .

From symmetry the area of GCE is also $x$ , so the area of EFGC is $2x$ .

The area of BEF is $2x$ as it has the same height and double base compared to CEF.

Note that BGC = BEF + EFGC so has area of $2x +2x = 4x$ , but obviously it is one sixth of the big square so its is area is 1.5 square centimetres.

Hence $4x=1.5$ so $2x = 0.75$ .

Since FC=GC and by symmetry of the square, EFCG is a kite. Note EDAB has the same internal angles, so EFCG and EDAB are similar.

Since DA/FC=3 and AE+EC= $3\sqrt{2}$ , we find AE= $\frac{9}{4}\sqrt{2}$ and EC= $\frac{3}{4}\sqrt{2}$ .

Then the area of ECG is $\frac{1}{2}\cdot1\cdot\frac{3\sqrt{2}}{4}\sin\left(\frac{\pi}{4}\right)=\frac{3}{8}$ ,

so the area of kite EFCG is $2\cdot \frac{3}{8}=\frac{3}{4}=\boxed{0.75}$ .