Need help

Break it down to shapes that you know.

• MNPQ square

• MQ and a corner forms a triangle

• NP and a corner forms another triangle

Now, both triangles have base and height as s/2. So their areas would be \frac{s^2}{8} each.

Next, you can use the Pythagoras theorem to find the side of the square MNPQ and find it’s area. Finally replace s^2 with A (given) and add up all the areas.

An alternative method would be to divide the shaded region into the parts I’ve mentioned before. And you can deduce that the inner square has half the area of the outer one, so that’s \frac{A}{2} . Similarly you can see that the rest half of the area is shared among 4 symmetrical triangles out of which we only want 2. So that’s \frac{A}{4}. Add them up to get \frac{3A}{4}.

Thanks