Let's say we have a perfect sphere with surface area equal to 1. In the following diagram (not to scale), I need to calculate the latitudes of parallels A through G (based on the following specifications), but I'm having no luck.
A is the north pole (90°N), and G is the equator (0°); I'm trying to solve for the latitudes of B, C, D, E, and F.
The spherical cap centered around A and bounded by B has surface area 0.015625 (1.5625% of the sphere)
The spherical segment bounded by B and C has surface area 0.015625 (1.5625% of the sphere)
The spherical segment bounded by C and D has surface area 0.03125 (3.125% of the sphere)
The spherical segment bounded by D and E has surface area 0.0625 (6.25% of the sphere)
The spherical segment bounded by E and F has surface area 0.125 (12.5% of the sphere)
The spherical segment bounded by F and G has surface area 0.25 (25% of the sphere)
As such, the total surface area of the cap and all the segments is 0.5 (50% of the sphere). How can I calculate latitude values for B, C, D, E, and F which provide segments of these sizes?
P.S.: If you don't know how to do this calculation, that's fine! That makes you just like me! Just refrain from commenting in that case please.
The area $\Delta A$ of each spherical strip bounded by two lines of latitude is proportional to the change $\Delta z= \Delta (\sin \lambda)$ that measures displacement in the vertical $z$ direction along the North-South axis of the unit-radius sphere. (Here $\lambda$ is the latitude angle.)
Since you know the percentage values of $\Delta A$ you can solve for the changes in $\lambda$ as you travel down the sphere.
See also spherical bread crust area formula