circumscribed circles

carnahanad

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Jan 30, 2012
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3
Hello,

I'm working on a CANstruction project where we build artwork out of canned goods (google it if you have more questions about CANstruction). The build we are working on requires me to make numerous circles. The cans we are using have a set radius, r1. I'm trying to figure out an equation that would tell me the radius, r2, of the larger circle circumscribed by the smaller circles. Yes, I realize I could take the cans and lay them out in different patterns in an imperical method of finding the radius. I was hoping to avoid that and derive a formula that would help me determine different r2s for different numbers of cans.

Basically I have "x" number of cans of r1 and I wasnt to know what r2 would be for x1, x2, x3, xi...

I have a feeling it has to do with 360/x and the tangent points of the smaller circles. Every approach I take seems to leave me with two many variables or a triangle with all the angles figured but no sides.

The point of the exercise is to know what r2s I can use so that I have a nice uniform circle with no gaps between the small cans.

Thanks for the help!
Alex
 
I'm working on a CANstruction project where we build artwork out of canned goods (google it if you have more questions about CANstruction). The build we are working on requires me to make numerous circles. The cans we are using have a set radius, r1. I'm trying to figure out an equation that would tell me the radius, r2, of the larger circle circumscribed by the smaller circles. Yes, I realize I could take the cans and lay them out in different patterns in an imperical method of finding the radius. I was hoping to avoid that and derive a formula that would help me determine different r2s for different numbers of cans.

Basically I have "x" number of cans of r1 and I wasnt to know what r2 would be for x1, x2, x3, xi...

I have a feeling it has to do with 360/x and the tangent points of the smaller circles. Every approach I take seems to leave me with two many variables or a triangle with all the angles figured but no sides.

The point of the exercise is to know what r2s I can use so that I have a nice uniform circle with no gaps between the small cans.

If you are talking about packing circles within a circle (rather than using circles to form a circumference of a large circle), this has been an area of long study. You can find some results here:

http://hydra.nat.uni-magdeburg.de/packing/cci/

http://www2.stetson.edu/~efriedma/cirincir/

http://www.buddenbooks.com/jb/pack/circle/n11.htm

http://www.buddenbooks.com/jb/pack/circle/n13.htm

The last two sites show two solutions each for 11 and 13 "cans" packed within circles -- demonstrating the complexities of packing problems. I hope this helps.
 
If you are talking about packing circles within a circle (rather than using circles to form a circumference of a large circle), this has been an area of long study. You can find some results here:

http://hydra.nat.uni-magdeburg.de/packing/cci/

http://www2.stetson.edu/~efriedma/cirincir/

http://www.buddenbooks.com/jb/pack/circle/n11.htm

http://www.buddenbooks.com/jb/pack/circle/n13.htm

The last two sites show two solutions each for 11 and 13 "cans" packed within circles -- demonstrating the complexities of packing problems. I hope this helps.

Thanks, the packing info will come in handy, but it's not quite what I'm looking for. I'm just looking for continuous circumfrence. We can use leveling plates, such as 1/8" plywood, between levels. So I don't necessarily need to pack the inner part of the circle.

I think it will be easier to do the emperical method, but now I'm invested in finding an answer.
 
Bull rushed it.

So, I did some imperical testing in AUTOCAD. I arrayed a line "x" times where "x" is the number of small cans with a radius of unit 0.5. I then placed a circle between two arrayed lines using the "tan, tan, radius" command for circle. From there, I was able to measure from the center of the arrayed lines to a point that passes through the small circle to the edge of the small circle. This would be my radius of the large circle. In the table below are my results. So 1 can circumscribes 1 "large" circle, so the "large radius" is 1. 2 small cans circumscribe a circle with radius of 2*.5=1.0, etc... If you check the differences between the different iterations of "large radius", after about 7 or 8 cans, the difference is pretty much the same. If you graph the small cans vs large radius starting with 3 cans, you get a trend line with y=0.1555x+0.5739 and this has an R2 value of 0.9998. It's not the prettiest way to do it, but I think it shows that there is a relationship that may be fairly simple. I included a picture of what I'm trying to do at the bottom, this instance is for 13 cans. So, from this equation I can get pretty close to knowing how big a circle I can make with "x" number of cans. I'd still like to know if anybody else comes up with any other ideas.
Small cans
degrees
large Radius
1
360
0.5
2
180
1
3
120
1.076823
4
90
1.207031
5
72
1.35026
6
60
1.5
7
54.42857
1.652344
8
45
1.80599
9
40
1.96224
10
36
2.11849
11
32.72727
2.27474
12
30
2.432292
13
27.69231
2.589844
14
25.71429
2.747396
15
24
2.904948
16
22.5
3.0625
17
21.17647
3.221354
18
20
3.378906
19
18.94737
3.53776
20
18
3.696615

Capture.jpg
 
wjm11's post above (referring to "using circles to form a circumference of a large circle") seems to be exactly what I'm looking for, but unless I'm reading something wrong I don't see how the calculation is explained.
num circles inside circumscribed circle.jpg

Hopefully the included drawing will give some idea of what I'm trying to do. Basically, if the width and height of the outer oval are known, and the diameter of the smaller circles are known, how many would fit around the inside of a larger oval that circumscribes those smaller circles? There's got to be an easier way to figure this out than by using GIMP (the poor man's Photoshop) to draw the large oval of a known dimension (in my case I scaled it to 10 pixels = 1 mm), draw the one small circle, copy & paste the small circles around the edge so they're exactly tangent, then count by hand the number if circles around the edge.
My online searches, for the most part, have come up futile, in part because I don't know what to search for. I just felt I got lucky and was getting close when I happened across the post referenced above, and because this thread is relatively recent (sometimes I'll come across a forum post in which the last post was made 8 to 10 years ago).
If someone could point me to a formula that would help me calculate this, it would help immensely, as there are several such calculations I'd like to do (including small circle diameters of 3, 4, 5, 6, 8 and 10 mm and large circumscribed oval dimensions from 0.5 x 0.5 inches to 5 x 8 inches in 0.5-inch increments, with the maximum height:width ratio being 2:1, for example).
For example, in the above example the oval is 2.5 x 4 inches, each small circle is 3 mm, and I was able to fit 75 (not quite 76) around it. In another example using a 2" circle, I could fit 48 3mm circles, 36 4mm circles, 23 6mm circles, 16 8mm circles or 12 10mm circles such that the circle that circumscribed them had a diameter of 2 inches.

Just to clarify, I'm only interested in how many circles fit around the circumference of the larger one, not how many will fit inside the total area of the larger circle.
 
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