*Spherical cap
https://en.wikipedia.org/wiki/Spherical_cap
Bonus points for whoever calculates the ratio of the surface areas for the two spherical caps that contain half of the human population. (The circle's radius is ~4,000 km)
https://en.wikipedia.org/wiki/Spherical_cap
Bonus points for whoever calculates the ratio of the surface areas for the two spherical caps that contain half of the human population. (The circle's radius is ~4,000 km)
Hmpf, I went to all the trouble of deriving the surface area of a spherical cap (it's really easy, as surface integrals go), and it's listed right there in the wikipedia page. A = 2πrr(1 - cos φ), where φ is the angle between the center of the cap and the edge (measured from the center of the sphere).
Assumptions:
1. The "radius" of the cap is 4000 km.
2. The "radius" of the cap is actually an arclength along a great circle of a sphere (the earth).
3. The radius (a real radius) of the earth is somewhere between 6300 and 6400 km.
The formula for arclength along a circle tells us that φ is somewhere between 40/63 and 40/64.
We can calculate the cap's percentage of the surface area of the entire sphere as (1 - cos φ) / 2. For an earth-radius of 6300km, this is 9.744%. For 6400km, this is 9.452%. For the smaller earth, the ratio of the cap's outside to its inside is 9.26:1; for the larger earth, it is 9.58:1.
Rounding to two significant figures, like we used for the size of the earth, the ratio should be between 9.3:1 and 9.6:1.
The ratio isn't really very informative, because the cap was drawn around where a bunch of people live, and "everything else" wasn't. This has artifactually put most of the ocean, where people cannot live, into "everything else". "Everything else" should probably be significantly deflated to adjust for this. The article messes up a similar point:
> The population density of Greenland, for example, is just 0.1/sq. mi—that is, one person living on every ten square miles of rock and ice. But it's a lot easier to find company in Manila, which is literally one million times as crowded: 107,000 Filipinos per square mile.
The population density of Greenland is mostly zero with some spikes. There's not a literal tenth of a person every square mile -- those people would all be dead. The population density in areas where local population density is more than zero is much higher. Manila is not actually one million times as dense.
I assume we are simplifying to an oblate spheroid and not worrying about topography and such.
Also I think this means technically its a Spheroidal Cap?
https://en.wikipedia.org/wiki/Coastline_paradox
So, yes, good point. Let's start with assuming a perfect sphere, then oblate spheroid, then topography, then only land (not water), and however crazy this thread gets with whatever else we haven't thought of :)