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Pearl Bracelet

If you have a pearl bracelet, what is the ratio of the radius of the pearls to the radius of the bracelet? Clearly it depends on the number of pearls in the bracelet, but what is the relationship?
We’ll make some assumptions:
  • The pearls are perfectly spherical.
  • The pearls touch each other with no gaps.
  • There is no clasp/hasp.
  • The band holding the pearls goes through the center of each pearl.
If you think about it, the center of each pearl forms is the vertex of a polygon on order n. We can draw a radius of the bracelet R to the center of each pearl, and make a right-triangle using the radius of the pearl P as the hypotenuse. The sine of the angle β is simply the ratio of the P/R.
We also know that there are 2n lots of β around the circle, which has radians, so β = 2π/2n. Combining these, sin(π/n) = P/R, and this gives us our answer.

Try it out

Below is a little app that allows you to experiment with adjusting the bracelet. Using the slider at the bottom you can adjust the number of pearls in the bracelet. In this animation, the radius of the bracelet is held constant at 1.0

Fixed Size Wrist

Of course, in real life, a person’s wrist is the constant dimension so in this variant there is a (totally anatomically correct) fixed circular wrist and adjusting the number of pearls in the bracelet adjusts the radius of the holding band so that the pearls fit around correctly.