For example, if the intersectional area between two circles was a square cm. Is it possible to say what the intersectional volume be if those same circles were spheres as a function of a?

To clarify : After solving for the overlap between two circles (see: https://i.imgur.com/uq9SV2d.png), you will know the area of each dome, a = total intersectional area, a1, being the half belonging to circle 1, radius 1, and a2 being the half belonging to circle 2.

Knowing the r1, and a1, how do you find the volume of a1 if the circle was a sphere? Basically like spinning around an arc into a dome.

Sorry, I'm not good at explaining.

Basically, if:

r1 = 10cm

a1 = area of the cap (overlapping area/2 if both r1 and r1 are the same) = 4 cm^2

What are the steps to find volume.a1 cm^3 given r1 = 10cm, a1 = 4cm^2? (or any other value for r1). The reason I want to know is I was thinking about the problem when trying to solve an issue with aproximating acceleration due to gravity, when two planets are overlapping the forces for that mass that's overlapping should cancel each other out as far as acceleration of the two planets is concerned.

By the way, how does that "put on hold thing" work? I tried editing the question. Is something supposed to happen?


closed as unclear what you're asking by Namaste, zz20s, José Carlos Santos, steven gregory, Jack D'Aurizio Apr 3 '18 at 22:58

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  • $\begingroup$ Are you familiar with methods of calculus? Or perhaps geometry $\endgroup$ – Triatticus Apr 3 '18 at 19:32
  • $\begingroup$ I haven't studied it before, except a little on my own. But I'm open to learning it now. The goal is to translate it into code, and since I already figured out how to calculate the area of intersection between two circles, I was hoping I could just use that when making calculations in 3D instead of redoing everything. $\endgroup$ – Bodhi1 Apr 3 '18 at 19:42
  • $\begingroup$ My suggestion is to look at some information on solids of revolution, that is what you might need for this $\endgroup$ – Triatticus Apr 3 '18 at 20:01
  • $\begingroup$ Yeah, it seems to me that after you know the area of the cap and the radius, you should have everything you need to turn the arc into a dome, but I can't figure out what steps to get there are exactly. $\endgroup$ – Bodhi1 Apr 3 '18 at 23:33
  • $\begingroup$ What can I do about the put on hold status? Is it still possible to get an answer? $\endgroup$ – Bodhi1 Apr 4 '18 at 14:38

Firstly, do not denote areas with N.

Start with simple. We have two intersecting circles with same radius and two intersecting spheres with the same radius $r$. The distance between centers of circles is equal to the distance between centers of spheres and it is equal to $d$. Calculating area of intersection of circles is easy. Calculate the area of sector segment and multiply by 2. And we have $2 \times (\frac{\alpha r^2}{2} - \frac{r^2 \sin \alpha}{2}) = (\alpha - \sin \alpha)r^2$. I hope you know what $\alpha$ is, and you can simplify $\sin \alpha$ in terms of $r$ and $d$.

Next thing is intersection of circles... Let A and B be (and bee $:)$ ) intersections and $O_1$, $O_2$ be centers. Calculate $\angle AO_1B$ in terms of $r$ and $d$. ( use cosine law if you don't think of any other way ) Then calculate height of spherical cap, it is $r \times (1- \cos \frac{\alpha}{ 2})$. Then calculate volume of sector and cone, and there difference should give you volume of cap and multiply it by 2, you will get $\pi h^2 (6r - 2h)/3$ (if I am not wrong, just check it, and h is height of cap).

Now, show Volume in terms of Area. And you will get the function you need. If difficulty arises, just comment.

  • $\begingroup$ How about circles with different radius? For example: i.imgur.com/uq9SV2d.png d = 1, r1 = 1, r2 = .5, a = .351. I guess in this case you would have to take a1 and a2 separately (both halves of the overlap) and use the r1 and a1 to calculate the volume of the end cap, and r2 and a2 to calculate the volume of the endcap 2 and add them together. I guess I'm lost on the exact steps required to go from a1 and r1 to the volume.a1. Using those values would you mind walking me through it step by step? Thanks for your help! $\endgroup$ – Bodhi1 Apr 3 '18 at 22:49
  • $\begingroup$ You need to calculate height of caps for each sphere. The method is the same. For area, it is simple, sum of areas of two sectors minus sum of areas of two triangles. You have two radii, and it is enough information. $\endgroup$ – Tahir Imanov Apr 6 '18 at 20:24

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