Puzzling Quarter Circle Area Problem [Geometry]

Some friends of mine recently showed me this problem from a 7th grade geometry assignment. To be sure, it was a bonus problem, but still, I thought it was pretty cheeky of the teacher to include it. Subscribe: https://bit.ly/polymathematic | Enable ALL push notifications 🔔 While I'm sure the teacher did not expect students to use the area formula for a symmetric lens, it does come in handy here. If you have something like the football shaped region in this problem, you can use the formula: r²(θ – sin θ). I've previously covered that formula in this video, on an infamous GCSE problem: Solving the Hardest GCSE Maths Question (https://youtu.be/deVzzwvnJM0). The r is the radius of the circles that create the lens, and θ is how much of those circles are used (up to π radians or 180º). In this case, being that the football shaped region is enclosed in two quarter circles, we want to use a quarter of the circle itself, which is π/2 radians (the formula requires that we leave θ in radians). After you have that football-shaped region (or symmetric lens), the problem is like many other composite area problems. Figure out what needs to be added or subtracted and do it. In this case, we have a large quarter circle, two semi-circles to subtract away, and then the football-shaped region, which we have to add twice. Once to compensate for the overlap of the semi-circles, and once because the area we want obviously includes it. Again, I doubt the 7th grade math teacher who assigned this expected their students to do all that. If we're very clever, we can approach this a bit more efficiently and notice that the football-shaped region actually fits in nicely with the outer area to make this simple a quarter-circle minus an isosceles right triangle. Fortunately, both methods give us the same answer, so we can be quite confident that the area in question here equals 4π – 8. #compositearea #symmetriclens #quartercircle I did eventually remember the other video I mentioned: my most recent video on using a special interactive unit circle in @Desmos . You can check that out here: https://youtu.be/8f9XGdHWj4w. Follow Tim Ricchuiti: TikTok: https://www.tiktok.com/@polymathematic Twitter: https://twitter.com/poly_mathematic Instagram: https://www.instagram.com/polymathematicnet Reddit: https://www.reddit.com/user/polymath-matic Facebook: https://www.facebook.com/polymathematic Watch more Math Videos: Math Minis: https://www.youtube.com/playlist?list=PLrc8spN1b3jkQynJ5heNvSs72gCPj_hwj Math Minutes: https://www.youtube.com/playlist?list=PLrc8spN1b3jmVFYwHiuMzCYJ0y_yiwUge Number Sense: https://www.youtube.com/playlist?list=PLrc8spN1b3jksKkY_oarNFEFAHjrIV97U MATHCOUNTS: https://www.youtube.com/playlist?list=PLrc8spN1b3jlqDkZXtby9lnEaOBy7-ZY0