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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.

Descriptive image

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.

mapp3r
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  • Here's a MathJax tutorial :) – Shaun Apr 24 '23 at 23:35
  • Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Shaun Apr 24 '23 at 23:35
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    Hint: area of the spherical cap – Jean-Claude Arbaut Apr 24 '23 at 23:39
  • I'm not asking for a direct answer, and I don't have the height (h) or the cap's base radius (radius of B), so the Wikipedia article is out. Please ensure your suggestions are relevant. – mapp3r Apr 24 '23 at 23:42
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    @mapp3r You don't need the base radius, please read more carefully before complaining. – Jean-Claude Arbaut Apr 25 '23 at 00:04
  • You can deduce the radius of the sphere using the condition $4 \pi r^2 = 1$. Then you could proceed using calculus taking the double integral of an angle along the great circle starting at the pole and the second path of integration a $2 \pi$ rotation around the pole. Setting equal to the surface areas you have provided, solving for the first angle I specified is the answer. – Phillip Hamilton Apr 25 '23 at 00:05
  • @Phillip Hamilton, the phrase "taking the double integral of an angle along the great circle starting at the pole and the second path of integration a 2π rotation around the pole" doesn't make English sense, and as such, I'm having difficulty understanding what you're referring to. Could you please rephrase? – mapp3r Apr 25 '23 at 00:08
  • Have you had any calculus? – Phillip Hamilton Apr 25 '23 at 00:09
  • Or is this just regular HS geometry? – Phillip Hamilton Apr 25 '23 at 00:10
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    Using formulas from the wikipedia link in @Jean-ClaudeArbaut's comment, the latitude of $B$ is $\sin^{-1}(31/32)$, so that article is definitely not out. – ronno Apr 25 '23 at 06:20

1 Answers1

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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

MathFont
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  • I could possibly do so if I knew more about the proportionality of their relationship. What I'm reading is that delta A = k * delta Z, but without knowing what replaces k, I don't have enough information to do so. – mapp3r Apr 25 '23 at 00:12
  • Total area of sphere of radius $R$ is $A=4\pi R^2$ and then total $\Delta z= 2R$ – MathFont Apr 25 '23 at 01:46