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November 11

Is half a sphere below |30°|?

I've suspected this since 8th grade but have been too lazy to finish adding cos(29.75°)*360*0.5, cos(29.25°)*360*0.5, ... cos(0. 25°)*360*0.5 to see if that's convincingly close to 1/4th the square degrees. Are there any other latitude bands with short descriptions that contain exact parts of a sphere's area with short descriptions? like 1/√2th the area, 5/12ths, 1/eth, 1/πth, e/π, 2/2π, 1 radian of latitude north and south, 90° minus 1 radian north and south that kind of thing. Sagittarian Milky Way (talk) 20:55, 11 November 2016 (UTC)[reply]

It is half the area (I take it that your angle is measured from the equator like latitude). If you do the integral, you will find that the fractional area between -θ and +θ is simply sin(θ). --catslash (talk) 22:03, 11 November 2016 (UTC)[reply]

See Trigonometric constants expressed in real radicals for nice values of sin(θ). --catslash (talk) 22:05, 11 November 2016 (UTC)[reply]

A pleasing consequence of this result, is that the fractional area of a band of latitudes is just the fractional length of the corresponding interval along the axis of the sphere. If you consider an interval of fixed length, and slide it along the axis towards one of the poles, the reducing radius of the corresponding latitude band is exactly counterbalanced by its increasing angular width as its surface tilts away from parallel to the axis - so that its area remains constant. --catslash (talk) 22:21, 11 November 2016 (UTC)[reply]

That is very neat, thanks. Sagittarian Milky Way (talk) 22:52, 11 November 2016 (UTC)[reply]

So if I knew this in 8th grade I could've just used the polar to Cartesian button on my calculator, lol. Sagittarian Milky Way (talk) 23:05, 11 November 2016 (UTC)[reply]

It's a theorem of Archimedes, by the way. See [1] for a pretty use of it. —Tamfang (talk) 18:17, 12 November 2016 (UTC)[reply]