Area by latitude

Area by latitude from the pole, an angle. Make rows out from the initial one million km^2, each subsequent row a whole number of million km^2; round up and find a new height.

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N Rows, 1000 km/row

Area at each row.

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 Area of a sphere = 

A

=

4

π

r

2

Area between the latitude, or 

A = 2*pi*r^2*(1-cos(theta))

1 - cos(theta) = 1 m. km^2 / (2*pi*r^2)

theta = arcos(1 - 10^6 km^2 / (2*pi*r^2) ), r = circ/(2*pi) = 40,000/2/pi = 6366.2 km

theta = arccos(1 - 10^6/2/pi/6366.2/6366.2) = arccos(1 - 0.003026988008686) = arccos(0.996073011991313) = 0.080877478519613

0.080877478519613/pi*20,000 km = 514.9 km

One million km^2

Points within each square.

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One million km^2 bits. Starting with a polar triangle fan. 

Diamond quads again...

These diamond quads...

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Go with a million km^2, like usual..

π*r^2, r = 1/π^0.5, radius of ~560 km