Formula to get the diameter of multiple circles to surround a larger one

February 13, 2016 8:35 AM

I am doing an art project that requires larger circles to be encircled by smaller circles. Each small circle must touch its larger one and its immediate small neighbors. The small circles must all be the same diameter. I would like to specify the diameter of the large circle and calculate the diameter and number of small circles required to accomplish the desired outcome. My geometry skills have eroded over these many years...help!
posted by txmon to Grab Bag (12 answers total)

Best answer: You're envisioning something like a quarter (large coin) surrounded by a ring of dimes (small coins), yes? Here are a couple of links that work that through -

-Mathematical Musings blog entry on circles surrounding circles

-Math Forum answer about finding the radius of small outer circles surrounding a larger circle
posted by LobsterMitten at 9:23 AM on February 13, 2016 [1 favorite]

Best answer: OK, so I'm going to think this one out loud, because it looks like a fun puzzle. Say you've got a circle 13cm in radius which you want to surround with 17 circles.

The diameter of the big circle is 2×13π, and the centres of the 17 circles around it make up a regular 17-gon with a side length equal to the diameter of the smaller circles.

We want to find the circumradius of the 17-gon which fits this, because when we subtract the radius of the inside circle from the circumradius, that gives us the radius of the small circles.

The circumradius is (s/2) + 13, and it's also s/(2 sin (π/17)). So we have (s/2) + 13 = s/(2 sin (π/17)). Copy and paste that into Wolfram Alpha, and you get 5.85cm.

So to surround a circle of 26cm diameter with 17 equal size circles, the surrounding circles need to have a diameter of 5.85 cm, with their centre on circle of 26cm + 5.85cm diameter: 31.85 cm.

Substitute other numbers in for 13 and 17, as appropriate.
posted by ambrosen at 9:28 AM on February 13, 2016

I think you need to add another constant to this equation. For any given circle, you could have as few as three 'surrounding' circles (though at that size they are obviously larger than the inner circle), and the upper bound would in theory be limitless given smaller and smaller surrounding circles.

On preview, what LobsterMitten said, keeping in mind that those formulas again require you to have a set number of surrounding circles in mind to determine what size they should be. If that's still a variable you should be able to extrapolate two options if your main decision point is not necessarily how many circles you have but approximately how large you want that outer ring to be.
posted by SquidLips at 9:31 AM on February 13, 2016

Bah! Curse you, lobstermitten. But at least I had a verified correct answer (1.42) to check mine against.
posted by ambrosen at 9:33 AM on February 13, 2016

A rough solution is: r_small = (π*r_large)/(n–π) (where n is the number of small circles).

The exact solution is trickier (I haven't figured it out yet) but it will give you a minimum n of 3.
posted by spacewrench at 9:36 AM on February 13, 2016

Best answer: I agree that without a chosen number of smaller circles, this problem is hard. It could work with ANY number of smaller circles.

If you have R and r as the respective radii, and you want N circles evenly around, then you need the following to be true:

360 = N*2*arcsin(r/(r+R))

(link to work)
posted by Wulfhere at 9:40 AM on February 13, 2016 [1 favorite]

Best answer: Here's a visualization that you can play with to get an idea of how it'll look like.

You can move the sliders or enter a specific radius for each circle.
posted by Wulfhere at 9:53 AM on February 13, 2016 [5 favorites]

For everyone playing along at home, here is an image you can print out and use to try to figure out what's going on.

(Wulfhere, if you were able to build that visualization in the same time it took me to draw an SVG picture, then ... wow!)
posted by spacewrench at 10:15 AM on February 13, 2016

oops, I fail at Imgur. Here is that link...
posted by spacewrench at 10:27 AM on February 13, 2016

I created a desmos graph (click here) that allows you to adjust the radius of the outer circle and the number of circles, and it calculates the necessary radius of the inner circles, and draws them.
posted by Salvor Hardin at 11:40 AM on February 13, 2016

We know that 6 identical circles will surround and touch a 7th circle identical to the six.

Therefore you need at least 7 surrounding circles to meet the criterion that they be smaller than the central circle.

If we set the radius of the inner circle to 1 for convenience, and label the unknown radius of the outer circles 'r', a little trig shows that

r/(1+r) = sin(2π/2n) = sin(π/n)

where n is the number of outer circles.

Rearranging, we get

r = sin(π/n)/(1-sin(π/n))

Checking this for the case n=6, where we know all 7 circles are identical and have radius 1, we get

r = sin(π/6)/(1-sin(π/6)) = (1/2)/(1-1/2) = 1

Which checks out.
posted by jamjam at 11:42 AM on February 13, 2016

Here is a general formula:

For any inner circle of radius r, the radius of n outer circles surrounding the inner circle have the radius, x, given by:

x = r/[csc(pi/n) - 1]

Hint: You can derive this formula using the chord length of a reference circle passing through the centers of the outer circles.
posted by incolorinred at 9:39 PM on February 14, 2016