How Do You Calculate the Area Between Two Overlapping Circles?

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    Multivariable
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Discussion Overview

The discussion revolves around calculating the area between two overlapping circles as one circle is shifted vertically over time. Participants explore geometric and integral approaches to determine the area of overlap at various positions of the circles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant describes the problem of finding the area eclipsed by one circle over another as the second circle is shifted vertically, providing the equations for both circles.
  • Another participant suggests that the overlap can be calculated using plane geometry, indicating that the overlap consists of sectors of the circles minus triangles.
  • A participant questions whether plane geometry can be used throughout the entire shifting process, rephrasing the question to seek the total area between the circles as the second circle moves from t=0 to t=200.
  • One participant proposes a method involving the radius of the circles and the length of the line connecting the overlap points, providing a formula for the area of the triangle and the sector of the overlap.
  • The same participant suggests that the final area calculation involves doubling the area derived from the sectors and triangles, while advising others to verify their calculations for understanding.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to calculate the area between the circles, with differing approaches and interpretations of the problem presented.

Contextual Notes

Some assumptions regarding the geometry of the circles and the nature of the overlap are not fully explored, and there may be unresolved mathematical steps in the proposed calculations.

balrog1212
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Hi,

I've been working on this problem for a few days and I'm pretty stuck. I'm working on how much area is eclipsed by some arbitrary planet by another over a given time. My question is if I have two circles one given by (y-100)^2+x^2 = 100^2 and the other given by (y+100)^2+x^2 = 100^2, how much area will be between those two curves as the second circle is shifted up until it is directly on top of the other circle?

my attempt to do this was to make the first circle (y-100)^2+x^2 = 100^2 and the second circle (y+100-t)^2+x^2 = 100^2. I would then want to find the area between those two curves as t goes from 0 to 200.

My ability to take integrals with multiple variables is poor but I will be taking calc 3 next semester so that should help.

any information on how to set this up or general help would be greatly appreciated.

Thanks
 
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You should be able to do this with plane geometry. Overlap consists of two pieces of the circles, where each piece is a sector minus a triangle.
 
Could I just use plane geometry to find the area between the two circles through the whole process of t going from 0 to 200?

I guess I could rephrase my question as "If I wanted to find the area between the two circles as t goes from 0 to 200 what would it be?"

at t=0 the area would be zero, at t=200 the area would be 10,000pi

so the total area I want to find would be (area at t=0+...+area at t=50+...+area at t=100+...+area at t=150+...+area at t=200) with every area in between those accounted for too.
 
The basic calculation gets one half, so at the end it is doubled.

Let r = radius of circle, s = t/2 (makes life easier for me). Consider line (L) connecting the two points where the circles overlap. The line from circle center to L has length r-s.
L has length 2√(2rs-s2). Therefore area of triangle is:
A=(r-s)√(2rs-s2).

The arc of the circle overlap is B=2arccos(1 - s/r). Therefore the area of the sector is C=Br2/2.

So the area you want is 2(C - A).

I suggest you check my calculations, so you will understand what is going on.
 

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