Here's the answer to the geometry puzzle: 8-8π+4√(2)π, or about 0.6388. The answer for side length x is ½x²(4-4π+2√(2)π). There are a few ways to get to it, but my thought process was this:

By applying the Pythagorean Theorem to the side length of the square, I can get the length of its diagonal.

Half of that diagonal is the radius of the larger quarter-circles.

Given that radius, I can find the area of those quarter-circles.

I can get the radius of the little quarter-circles by subtracting the radius of the big quarter-circles from the side length of the square.

That gives me the area of those small quarter-circles.

I can get the area of the big square from its side length.

If I subtract the area of two big quarter-circles from the area of the square, I get two of the red sections plus two of the small quarter-circles. (If it's not clear why, sketch it and shade out two big quarter-circles which are opposite each other.)

If I subtract the area of the two small quarter-circles from that, I have the area of two of the red parts.

I can double that to get all four.

The problem itself isn't that hard, but mashing all the numbers together required a lot of simplifying equations. After I caught myself about to screw up a sign, I was certain I must've made another mistake elsewhere and not seen it yet, but nope, the person I got the problem from confirmed it.