How Does One Deduce Area of Intersection for Three Cirlces?

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Homework Help Overview

The discussion revolves around determining the area of intersection for three circles, each with the same radius. The original poster expresses uncertainty about how to approach the problem, particularly in relation to the geometry involved with the circles and their centers.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the properties of triangle ABC formed by the centers of the circles, noting it is equilateral with 60° angles. There is exploration of the area of the triangle and how it relates to the overall area of intersection. Questions arise about how to calculate the areas of the circular segments outside the triangle and the relevance of circular sectors.

Discussion Status

The conversation is ongoing, with participants providing insights into the geometry of the problem and questioning assumptions about the areas involved. Some guidance has been offered regarding the calculation of areas related to circular sectors, but no consensus has been reached on the final approach.

Contextual Notes

Participants are working under the constraints of a homework assignment, which may limit the information they can use or the methods they can apply. There is an emphasis on understanding the relationships between the triangle and the circular segments.

cmkluza
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Homework Statement


upload_2015-3-26_16-41-20.png


Homework Equations


Most likely Acircle = πr2
Not sure if there's others.

The Attempt at a Solution


I'm not sure where to start, I've never seen a question of this sort. They all have the same radius, hence same area, and each point/center is r away from another, but I don't know how, if at all, that gets me closer to understanding.

Thanks for any help ahead of time!
 
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What can you say about triangle ABC?
 
SteamKing said:
What can you say about triangle ABC?
Well, I can see that triangle ABC must be equilateral, and so it must have three 60° angles. Is that what you're hinting at, or did I miss something? If yes, I'm still missing where to go next. The area of the triangle would be r2/2 and there would be the area of the small segments outside of the triangle left to find the entire area.
Thanks!
 
cmkluza said:
The area of the triangle would be r2/2

Is that your final answer for the area of triangle ABC?

Maybe you should make a sketch before stating this area flat out.
 
SteamKing said:
Is that your final answer for the area of triangle ABC?

Maybe you should make a sketch before stating this area flat out.
Sorry, that was an obvious error on my part, not sure what I was thinking. Area of the triangle should be 1/2 * r * √(r2 - r2/4) = r/2 * √[(4r2 - r2)/4] = r/2 * √(3r2/4) = r/2 * r/2 * √(3) = (r/2)2 * √(3) So, where do I go from here to integrate the π into my answer?

Thank you very much for your help thus far!
 
cmkluza said:
Sorry, that was an obvious error on my part, not sure what I was thinking. Area of the triangle should be 1/2 * r * √(r2 - r2/4) = r/2 * √[(4r2 - r2)/4] = r/2 * √(3r2/4) = r/2 * r/2 * √(3) = (r/2)2 * √(3) So, where do I go from here to integrate the π into my answer?

Thank you very much for your help thus far!

The triangle ABC covers most of the area of overlap amongst the three circles. Think about what it doesn't cover: namely the area outside the triangle ABC but inside the circular arcs AB, BC, and CA. How would you figure these areas?

Hint: think about what the area is of a circular sector, for example one centered at point A and including arc BC. Since you know what the central angle of the arc BC is, what would be the area of this circular sector? Given that the area of the whole circle is πr2 and the central angle for the whole circle is 2π radians, what would be the area of a portion of the circle which had a central angle of θ radians?
 

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