How Does One Deduce Area of Intersection for Three Cirlces?

[cmkluza]

Homework Statement

Homework Equations

Most likely A_circle = 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!

 

[SteamKing]

What can you say about triangle ABC?

 

[cmkluza]

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!

 

[SteamKing]

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.

 

[cmkluza]

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!

 

[SteamKing]

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?