I didn't proceed that way, but I imagine this may work too. I'll post the answer tomorrow.

south_korea_ln

It might help to assemble four quarter circles into a full one:

Let's say the doubly shaded right triangles have shorter sides 

We have to subtract the four white rectangles with area 

Further add the doubly shaded triangle areas 

Then the answer is obviously a fourth of that.

That's how far I got, but I couldn't get 

science

[Not-so-daily puzzle] A fun geometry puzzle

It might help to assemble four quarter circles into a full one:

![circle](https://i.ibb.co/mzBgwKd/circle.png)

Let's say the doubly shaded right triangles have shorter sides $$x \gt y$$.

The outer square's side length is $$2x+12$$, area $$4x^2+48x+144$$.

We have to subtract the four white rectangles with area $$1+x$$ each.

Further add the doubly shaded triangle areas $$2xy$$ (they count twice).

Then the answer is obviously a fourth of that.

That's how far I got, but I couldn't get $$x$$ and $$y$$.