Date modified: Mon 2023-09-18 . 09:01 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Extra credit problem: Find the area of something geometric. Many many years ago I came across and read this book called "One, two, three...Infinity" by George Gamow, it's pretty interesting, and I recommend you a read. But what looked cool to me at the time was its book cover: ![[1 teaching/smc-summer-2023-math-8/problems/week-7/---files/Pasted image 20230802101626.png]] What would be the area of the black circular caps? Stating it more precisely: Consider an outer unit square (side lengths are all 1), and an circle inscribed in it (meaning the circle just touches all sides of the square). Now in this circle, we inscribe another square (meaning the corner of the square just touches the circle). We repeat this process, inscribing a circle, then a square, and so on. Infinitely many times. Find the area of all infinitely many circular cap regions, shaded as shown: ![[1 teaching/smc-summer-2023-math-8/problems/week-7/---files/extra-credit-problem-area 2023-08-02 10.07.08.excalidraw.svg]] %%[[1 teaching/smc-summer-2023-math-8/problems/week-7/---files/extra-credit-problem-area 2023-08-02 10.07.08.excalidraw|🖋 Edit in Excalidraw]], and the [[smc-summer-2023-math-8/problems/week-7/---files/extra-credit-problem-area 2023-08-02 10.07.08.excalidraw.dark.svg|dark exported image]]%% Show your work and your answer should be an exact expression of the area. Hint: Wouldn't you say this is quite **geometric**? ////