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June 3rd, 2019, 06:49 AM   #1
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Cool Please explain to me in simple terms what these circle intersections are all about.

So, say you got 4 circles intersecting this way:


Now, I am looking for two things:
  • A proof that each part of the circle which is in an intersection is 1/4 the size of the whole circle's circumference
  • The exact area of the non-shaded region.

Now, in my search for finding the answer to this, I stumbled upon this article: http://mathworld.wolfram.com/Circle-...ersection.html. The only problem? I have no idea what this article is trying to say, and how it can help me. I did find the equation to get the area of the shaded region (it's $\displaystyle A=2(\pi-2)$) which I can use in Part 2 but I still don't understand how the solution got to there, and how to do Part 1. Please help me in learning what is trying to be said here in simpler terms! Thanks!

Last edited by skipjack; June 3rd, 2019 at 11:49 AM.
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June 3rd, 2019, 11:52 AM   #2
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The article starts by considering circles of (possibly) different size, but later considers circles of the same size without making that clear.

Which steps were you unable to follow? Are you unfamiliar with the Cartesian equation of a circle?
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June 3rd, 2019, 01:39 PM   #3
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For part 1, assuming the circles are all the same, let $r$ be the radius of the circles:

I'll just apply this general idea to the top and right circle since this idea can be applied to the other circles as well.

Draw a horizontal line from the center of the diagram to the center of the right circle, this distance is $r$.
Draw a vertical line from the center of the right circle to the intersection of the right and top circles, this distance is also $r$.
Call these lines $A$ and $B$ respectively

Draw a vertical line from the center of the diagram to the center of the top circle, this distance is $r$.
Draw a horizontal line from the center of the top circle to the intersection of the top and right circles, this distance must also be $r$.
Call these lines $C$ and $D$ respectively.

The lines $A$, $B$, $C$, and $D$ are all length $r$ and form a square. Therefore the angles between $A$ and $B$ as well as between $C$ and $D$ are $90$ degrees.

Since $90$ is a fourth of $360$ , the arcs formed by these angles must be a fourth of the total circumference.
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June 3rd, 2019, 01:45 PM   #4
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Quote:
Originally Posted by skipjack View Post
The article starts by considering circles of (possibly) different size, but later considers circles of the same size without making that clear.

Which steps were you unable to follow? Are you unfamiliar with the Cartesian equation of a circle?
Yes, I am unfamiliar with the cartesian equation of a circle.
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June 3rd, 2019, 04:03 PM   #5
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This article provides a very brief introduction. Also, you need to be aware of this theorem so as to understand why the Cartesian equation for a circle is what it is.
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June 5th, 2019, 10:39 PM   #6
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For the second part just make a sqare inside a circle and subtract the area of square from the circle and analyse.
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