https://www.varsitytutors.com/hotmath/hotmath_help/topics/surface-area-of-a-sphere
This one says the area of a ball is the same with the are of cylinder surrounding it.
Why?
How did Archimedes figure out that the area of ball is the same with the area of cylinder surrounding it
area
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Best Answer
If you are looking for something intuitive back then, there is one way. If we take a rope of fixed (but minimal thickness), it can act as an integrable element (like discs of rings used in calculus).
1. Now start wrapping the rope around the sphere. Cut-off the rope at the length where it has covered the whole sphere. This is A1.
2.
2a. Now again take another rope. Wrap it around the cylinder along it's height. Cut off at the length where the whole cylinder gets wrapped. This is B1.
2b. Take another piece of rope. From the centre of the circular base/top of the cylinder, start coiling it along the boundary until the whole area of the circle is covered. Cut at this lenght. This is B2. Do the same for the top/base. This will be B3.
Now when you compare the length of A1 vs the length B1+B2+B3 ,under the errors of physical measurement, they will be of same length. Multiply the length of the rope with it's mean diameter, and you will have nearly equal areas too! (since the area of that elemental strip will be it's length into it's width).
Here's the pic for help in visualization:
It's probably one of the ways early mathematicians could have done it without any proper knowledge of trigonometry or calculus.