I need to apply the formula for the area of the section of a sphere. I work with zenith and azimuth angles, and I need to find the height of my sphere section depending on my zenith (theta) angle. I know my sphere radius R, and the two angles that will define the section I'm looking for. I can reason in 2 dimensions for this calculation.
Given two zenith angles, what is the height of the circle section that emerges?
Firstly, I need to find the height of the "cap" of my circle, between theta=0 (vertical angle), and theta1.
I'm having trouble putting this in words, so here are two diagrams. I am looking for h each time:
- for the specific case of the "cap" of the circle:
- for two different angles, theta1 and theta2:
Being an outsider to this kind of calculations, I tried to reason with what I already know. I was thinking of finding h using Pythagoras theorem, knowing that h is R - something. But I'm afraid my brain is not wrinkled enough for this! I found this answer that could be a pointer, but OP's question doesn't reason in terms of angles, but of distances.
By using trig in the right-angled triangle, the distance between the cutting plane and the origin is $R \cos \theta$. So in the special case, the height of the cap is $$h = R-R\cos \theta = R(1-\cos \theta)$$ and in the case of the slice you have $$h = R\cos \theta_1-R\cos \theta_2 = R(\cos \theta_1-\cos \theta_2 )$$ assuming that $0\le \theta_1 \le \theta_2 \le \pi$. Note that the special case is simply the normal case with $\theta_1=0$.