Chords of a Unit Circle

Take N equidistant points on the . Pick one of those points, then draw from it to all points on the circle, as in Figure 1. What is the product of lengths of those chords?Surprise: for N points, the product is just N. Pretty cool, huh?

Presentation Suggestions:
Do simple cases first and draw pictures: for N=2, there is just one chord (the diameter of the circle, length 2). For N=3, there are two chords (both of length Sqrt[3], product=3). For N=4, there are 3 chords (one diameter of length 2, two of length Sqrt[2], product=4). Have people conjecture the answer for themselves!

The Math Behind the Fact:
If you know about , you can prove this by letting the points of the circle be given by roots of unity, and letting z be any point (not necessarily the unit disc) and looking at the product of lengths of the chords from z to all N points on the circle. Divide this expression by (z-1), and take the limit as z goes to 1, and voila!, you will get N.

If you liked this Fun Fact, now see what happens with Chords of an .

How to Cite this Page: 
Su, Francis E., et al. “Chords of a Unit Circle.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Francis Su

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