Steins Gate With Dat Math

Connect adjacent midpoints of the square to make a diamond.

Note that the sum of the heights of opposite triangles formed by the diamond and the 4 line segments is equal to the side length of the diamond; thus, the sum of areas of opposite triangles is half that of the diamond.

This means that the sum of one pair of opposite triangles is the same as the sum of the other pair.

Now just note that the original diagram adds a 45-45-90 triangle to each triangle, thus the property remains that the sum of the areas of the pairs of opposite shapes is equal.

So $A+20=16+32$ or $A=\boxed{28}$ .