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I found this puzzle here. (It's labeled "crossed cylinders".) Here's the description:

Two cylinders of equal radius are intersected at right angles as shown at left. Find the volume of the intersection between the two cylinders, without using calculus! A 3D picture of the intersection is shown at right.

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Hint (medium hint - exactly which high school formulae you need): 1) Area of circle = pi * radius2, and 2) Volume of sphere = (4/3) * pi * radius3

Note: Solved by the mathematician Archimedes (287 B.C. - 212 B.C.), waaay before calculus came around!!

Please tell me how to solve this puzzle! Is there a way to do this without setting up a Riemann sum and finding a limit, essentially evaluating an integral?

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up vote 10 down vote accepted

Compare the solid to a sphere of the same size. The slices of the solid are squares to the sphere's circles, so the ratio of areas is always $4/\pi$. The volumes must therefore be in the same ratio, giving $(4/\pi) \cdot (4/3)\pi r^3 = (16/3) r^3$.

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Wow, that's brilliant, thank you! –  littleO Nov 15 '12 at 12:28
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