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3 Circles in a Circle

M

Minjin

Platinum Member
I was doing some reading on the DIY Audio forum in the Subwoofer section about sonosubs and it led me on a train of thought that ended thusly:

You have a circle of known radius and you want to fit THREE identical circles inside of it. Whats the largest they can be?

My drawings with squarish looking circles aren't sparking any ideas here. I'm going to think about it some more tomorrow after I get some desperately needed sleep but I thought I'd post this anyways.

Can it be solved geometrically? I have a feeling that this is pretty simple for a geometry student but its been about 14 years since I've had that class. 😉

Mark
 
Now, time to create a formula for the maximum radius of any whole number of circles above two that can inscribed in a larger circle.
 
Not sure you have given enough information in your OP. You could easily fit 3 identical circles that coincidentaly all fit on top of one another inside the circle with each having a diameter of x-.0000000........1 where x is the diameter of the original circle. Please be more precise. Can the circles overlap, share the same space, only fit adjacent to one another?
 
Three identical circles that kiss but don't otherwise occupy each other's space. I suppose I was remiss in not saying identical. I'm been google searching since I posted the OP and it looks like it might have something to do with Descartes Circle Theorem but I'm too tired to make any sense of it.

Mark
 
obviously the inner circles are arranged with their centers in a triangle, and the big circle fits around the outside. you people should be able to figure this out from the op, despite it not being specified
 
Originally posted by: iwantanewcomputer
obviously the inner circles are arranged with their centers in a triangle, and the big circle fits around the outside. you people should be able to figure this out from the op, despite it not being specified
Click to expand...

Obviously you don't have a clue about setting parameters for a particular problem.

 
Originally posted by: LASTGUY2GETPS2
Originally posted by: her209
(pi)(r^2) >= 3(pi)(x^2)

Solve for x where x is the largest value

x should be between 0 and r
Click to expand...

That exactly how I would've done it. Don't know for sure if is right though.
Click to expand...


That's not right because it doesn't take into consideration that it's a circle. If you can squish the little circles into like different forms it would make sense.
 
Originally posted by: Midlander
Originally posted by: CaesaR
Radius = 0.46*r
where "r" is the radius of bigger circle.
Click to expand...

This looks about right, but how did you get this equation?
Click to expand...

The radial lines from centre of bigger circle that are tangents to one smaller circle are separated by 120degrees. Now the line from centre of big circle to point where it touches the smaller one is s = x/tan(60) where x is the radius of smaller circle. the distance between the two centres (big and small) is l = sqrt (x^2 + s^2). the radius of the small circle
x = r - l

Edit: hope its clear cuz I'm just too bad at ms paint with touchpad
 
Edit:

This should answer your question.

Looking at the smallest triangle: Set its hypotenuse to 2. Its longer side is then sqrt(3). Note that this is the radius of the smaller circle. Note also that the radius of the larger circle is 2+sqrt(3).

You can go from there lmk if you have any issues.
 
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