3 Circles in a Circle

[Minjin]

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

 

[her209]

n/m

 

[Minjin]

Originally posted by: her209
n/m

Heh. I was just about to quote you and ask you what you were smoking.

Mark

 

[Leper Messiah]

I think I get what you're saying but...um...it would have alot to with arcs and tangents and crap like that.

 

[her209]

(pi)(r^2) >= 3(pi)(x^2)

Solve for x where x is the largest value

x should be between 0 and r

 

[daniel1113]

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.

 

[conehead433]

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?

 

[Specop 007]

Graph paper.

 

[Minjin]

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

 

[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

 

[SaturnX]

I'm assuming this would be what you're talking about?

circles.gif

--Mark

 

[conehead433]

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

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

 

[ArJuN]

Give me a second, I'm going to give this a shot.

 

[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

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

 

[Caesar]

Radius = 0.46*r
where "r" is the radius of bigger circle.

 

[Midlander]

Originally posted by: CaesaR
Radius = 0.46*r
where "r" is the radius of bigger circle.

This looks about right, but how did you get this equation?

 

[ArJuN]

Screw this, it's friday night. (aka, I can't do it)

 

[Ready]

I dunno, I end up with Big radius = 2.1547 small radius

 

[Minjin]

Originally posted by: SaturnX
I'm assuming this would be what you're talking about?

circles.gif

--Mark

Exactly.

Originally posted by: CaesaR
Radius = 0.46*r
where "r" is the radius of bigger circle.

Thanks, and how basically did you come up with that?

Mark

 

[ArJuN]

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

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

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.

 

[Caesar]

Originally posted by: Midlander

Originally posted by: CaesaR
Radius = 0.46*r
where "r" is the radius of bigger circle.

This looks about right, but how did you get this equation?

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

 

[Ready]

I ended up with the equation
Big radius R
small radius r

R = r[1+2cos(30) - tan(30)]

 

[zendari]

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.