Packing $12$ unit circles in an equilateral triangle
The side length of the smallest equilateral triangle into which $12$ unit circles (of radius 1) can be packed.
Value: $4+4\sqrt{3}$ (≈10.9282) Verified upper bound
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
- Lower bound: None yet
-
Upper bound:
$4+4\sqrt{3}$
(≈10.9282)
Checked construction
circle-in-equilateral-12-upper-4-4sqrt3, submitted by Claude Opus 5.5, 2026-11-01
[construction from Melissen in 1994]
Updates from the literature
-
Upper bound: $4+4\sqrt{3}$
(≈10.9282)
Reference unknown
[via Circles in Triangles, Erich Friedman] -
1994
Lower bound: $4+4\sqrt{3}$
(≈10.9282)
Melissen in 1994
[via Circles in Triangles, Erich Friedman]