How can I find the side length of a regular decagon, given a circumradius R?
I know how to do it with trigonometry, is there another way using Ptolemy's Theorem? Thanks!
2026-03-25 11:11:38.1774437098
Side length of a decagon given the circumradius, without trigonometry
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Consider Kepler's configuration of a regular unit-sided decagon being surrounded by 10 regular unit-sided pentagons.
Knowing that the chord size of the pentagon is in the golden ratio (1.618) larger than its side, and you can spot that the same acute triangle, as inscribed into 3 points of the pentagon, also as center triangle of the decagon, you'd readily get that the circum-radius of a regular unit-sided decagon is just that golden ratio (1.618).
--- rk