Solving the Circles

Occasionally, people come to me when they spot a problem they want to solve and think it might be possible with the application of mathematics.  This is really fun, because it lets me exercise that part of my brain and at least try to do some algebra from time to time.

Recently, a friend of mine came to me with this picture:

She was interested in how big the rings were, so that she could try to make something similar.  To help model what was going on, I made the following assumptions:

1. The jump rings can be arbitrarily chosen to fit around the width of the other rings, and their size can be chosen to give the item more play, but aren’t determined explicitly by the shape.
2. The medium ring (on the top) is slightly larger than could be surrounded by seven small rings (noted by the fact that the very top small ring is just slightly off-center to the left, and the bottom two rings are centered at the bottom.)
3. The large ring (on the bottom) could be surrounded by ten small rings (noted by the fact that the two small rings on the top align with the two small rings on the bottom of the design, splitting the ring in half.)

Now for the math part.

Suppose the radius of the small rings is X, the medium ring is Y, and the large ring is Z.  The lengths obey the following diagram:

Where, using the formula for the length of a chord of 1/7th of a circle of radius X+Y, we get:

2x = 2(x+y)sin(π/7)

Similarly, for the length of a chord of 1/10th of a circle of radius X+Z, we get:

2x = 2(x+z)sin(π/10)

Solving for Y and Z in terms of X, we get:

y = x/sin(π/7) – x
z = x/sin(π/10) – x

If we start with a small ring that is 1/4″ diameter, the medium ring has a diameter of 0.32619″ (or slightly larger) and the large ring to has a diameter of 0.55901″.  Rounding for convenience gives the medium ring a diameter of 3/8″ and the large ring a diameter of 9/16″.

If we start with a small ring of 3/8″ diameter, the medium ring has a diameter of 0.48928″ (or slightly larger) and the large ring has a diameter of 0.83832″.  Rounding for convenience gives the medium ring a diameter of 1/2″ and the large ring a diameter of 7/8″ or 13/16″.