# Finding area of circular lunes using elementary geometry

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?

• Related: "Quarter-circle and circle inscribed in square", "How to find the area of intersection of two circles using axiomatic geometry?", and probably more; I'm fairly certain I've seen the image from the screenshot here before. (Searching for things like this is why descriptive titles are important!) Of course, these don't address what went wrong with your approach, but cross-references can be helpful.
– Blue
Jul 31, 2023 at 19:04
• @Blue I'll check those references out, thanks. Jul 31, 2023 at 19:05
• Your 'area of the football region that intersects the circle' seems wrong because it incorrectly subtracts the extra (smallest) non overlapping thin shaped areas. I'll post it as answer, Jul 31, 2023 at 19:07
• Oh right, I overlooked that. Thanks a lot. That would also explain why my answer is too large. Jul 31, 2023 at 19:08