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January 4

Area of a slice of a sphere

I've just been trying to find the area of the curved surface of a slice of a sphere by integrating the figure of rotation. I'm getting the counterintuitive result that all slices of a sphere of the same thickness have the same area of the curved surface (obviously, the circular flat surface is not the same). Is that right? Is it related to the area of a sphere being equal to the area of the cylinder it inscribes? SpinningSpark 15:23, 4 January 2017 (UTC)[reply]

Yes. I don’t know there is any deeper reason beyond 'it’s true' and the maths you have already used. See Sphere#Surface area for some background.--JohnBlackburne^words_deeds 15:33, 4 January 2017 (UTC)[reply]

MathWorld leaves no indication either. Here is a paraphrase of the calculus that I find somewhat illuminating: let's say the sphere is of radius 1. The area of the slice between heights $z=\cos \theta$ and $z+dz$ is equal (for dz small enough yada yada) to the product of a circumference of radius $\sin \theta$ (the radius the slice has at $z$ ) i.e. $2\pi \sin \theta$ with a height equal to the length of the arc of great circle between $z$ and $z+dz$ , i.e. $dz/\sin \theta$ . Hocus pocus, the sines disappear; but it does not really tell us how it comes that the slice is inclined just enough to compensate for the fact that it has a smaller diameter. Tigraan^{Click here to contact me} 17:18, 4 January 2017 (UTC)[reply]

I was working with Pythagorians rather than trig functions, but I got a similar cancellation. It's magic whichever way you look at it. SpinningSpark 17:37, 4 January 2017 (UTC)[reply]

Yes, and if you like that you'll also like the Napkin ring problem. A hole is drilled going through the center of a sphere. The distance between the ends of the hole is 4 inches - what is the volume of the remainder of the sphere ((The width of the hole isn't given). Dmcq (talk) 16:43, 4 January 2017 (UTC)[reply]

This question came up only last November [1] --catslash (talk) 19:58, 4 January 2017 (UTC)[reply]

More directly: Wikipedia:Reference desk/Archives/Mathematics/2016 November 11. (It's the only question from that day, which is lucky as I don't know how to escape the ‘|’.) —Tamfang (talk) 07:19, 7 January 2017 (UTC)[reply]