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Math puzzle: The sum of each circles
By Jack Murtagh
Amanda Montañez
A reddish circle is inscribed wrong a bluish square. The statement leaves gaps successful nan square’s 4 corners, 2 of which are filled pinch smaller circles that conscionable hardly touch nan large reddish circle and nan 2 area sides of nan bluish square. This, successful turn, leaves 2 smaller gaps successful nan corners, which are filled pinch smaller circles, and truthful on, pinch ever smaller circles advertisement infinitum. The full sketch is inscribed wrong of a 1 × 1 grey square. What is nan full circumference of each nan circles?
The full circumference of each nan circles is π. The circumference of a circle is π times its diameter. So nan full circumference of galore circles pinch diameters d1, d2, d3,... is:
πd1 + πd2 + πd3 + ... = π(d1 + d2 + d3 + ...)
So if we tin find nan sum of each nan diameters, we tin multiply that worth by π, and we’re done. Because of nan symmetry, nan sizes of nan circles don’t alteration if we move immoderate to different corners:
Amanda Montañez
Because infinitely galore circles thin toward nan corners of nan bluish square, nan sum of nan circles’ diameters equals nan magnitude of nan diagonal of nan bluish quadrate (pictured arsenic a dashed line). This magnitude equals 1 because nan outer grey quadrate has broadside magnitude of 1.
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