Surfing shapes
Consider two shapes: (a) Cylinder with radius R and height H and (b) Stack of spheres, each with radius R. You stack up enough number of spheres so that the total stack height is H. (Truncate the top sphere, if needed, to match the height H.)
Question: Which of these two shapes have more curved surface area?
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1 solution
The area of the curve surface of a cylinder is circumference x height i.e 2πR.H.
The area of a sphere is 4πR^2.
Let's suppose H = 2nR, which means we could stack n spheres to get exactly H.
The area of the cylinder would be 2πR.2nR = 4nπR^2. The total area of n spheres is 4nπR^2 too. Both are equal.
This is also true if we have to truncate the sphere.
Archimedes showed that, for a truncated sphere which height is K and greatest radius is R, the area is 2πRK.
The area of the truncated sphere is the same as the curve area of the tangent cylinder with same height.
That's the reason why a sculpture of a sphere and a cylindrer was put on Archimedes' grave (Cicero said, according to Wikipedia).