Finding area of circular lunes using elementary geometry

Question

I tried to solve the geometry problem in the YouTube video "Killer Problem for Students in China" from MindYourDecisions. Here's a screenshot:

My Solution

Dividing the square into two right triangles of area $50$ each, the area of the football-shaped region is

$$2(25\pi-50)=50\pi-100\approx57.08$$

The area of each of the regions in the corners is

$$\frac{100-25\pi}{4}\approx5.365 $$

So the area of the football region that intersects the circle is $57.08-2(5.365)\approx46.35$, which means the area of the shaded regions is $25\pi-46.35\approx32.19$ square centimeters.

But the solutions given by the video are much more complex and the correct answer is about $29.276$ square centimeters.

I have reviewed my somewhat oversimplified solution, though, and cannot find the flaw. Could someone point it out to me?

Answer 1

Your 'area of the football region that intersects the circle' seems wrong because it incorrectly subtracts the extra (smallest) non overlapping thin shaped areas.