How Does One Deduce Area of Intersection for Three Cirlces?

In summary: The area of 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?The area of the triangle ABC covers most of the area of overlap amongst the three circles. However, it does not cover the area inside the circular arcs AB, BC, and CA. To find the areas inside the arcs, you would need to figure out the central angles for each one. For arc AB, the central angle is 2π radians. So the area of the sector would be π*(2π*r*cos(
  • #1
cmkluza
118
1

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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  • #2
What can you say about triangle ABC?
 
  • #3
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!
 
  • #4
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.
 
  • #5
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!
 
  • #6
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?
 

1. How do you find the area of intersection for three circles?

To find the area of intersection for three circles, you can use the formula A = r^2 * cos^-1((d^2 + r^2 - R^2)/(2dr)) - (d/2) * sqrt((-d+r+R)(d+r-R)(d-r+R)(d+r+R)), where A is the area of intersection, r is the radius of the circles, and d is the distance between the centers of the circles. This formula can also be used to find the area of intersection for any number of circles.

2. What is the significance of finding the area of intersection for three circles?

The area of intersection for three circles is useful in many applications, such as geometry, physics, and engineering. It can be used to determine the overlapping region of three objects, the amount of space that three objects share, or the common area of three events occurring simultaneously.

3. Can the area of intersection for three circles be negative?

No, the area of intersection for three circles cannot be negative. It represents the amount of space that is shared by all three circles, so it will always be a positive value.

4. Are there any other methods for finding the area of intersection for three circles?

Yes, there are other methods for finding the area of intersection for three circles, such as using numerical integration or approximating the area with a polygon. However, the formula mentioned in the first question is the most commonly used and accurate method.

5. How accurate is the formula for finding the area of intersection for three circles?

The formula for finding the area of intersection for three circles is very accurate, as long as the circles do not overlap completely or have a very small distance between their centers. In these cases, the formula may not give an accurate result and other methods may be needed.

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