Answer for the square puzzle
To get to the answer, first let’s visualize the quadrants a little differently. Draw a line (the red dotted ones here in the picture) from the point in the square to each of the vertices. So, the four quadrilaterals have been now divided into eight triangles. Here comes a twist – instead of thinking about the two triangles in any quadrilateral, think about any triangle and the other adjacent one from it. (other than the one in its own quadrilateral).
In the picture, I have colored them. The two greens, the two yellows etc. What can you say about the two green triangles? If you remember old geometry, they have the same base (half a side of the square) and same height. Thus the two green triangles have same areas. Similarly, the two yellows have same area. As do the two blue. And the two purple.
Now think about the quadrilateral with area 80 and that of the one with 40 (diagonally opposite). One green + one blue + one yellow + one purple. And that is 80+40=120.
Take the other two quadrilaterals. You will notice that they also add up to one green + one blue + one yellow + one purple. Which means the unknown quadrilateral has to be 120 – 60 = 60.
Thus the area of the square is 80+60+40+60=240 units !