Packing $9$ unit circles in an equilateral triangle
The side length of the smallest equilateral triangle into which $9$ unit circles (of radius 1) can be packed.
Value: $6+2\sqrt{3}$ (≈9.464102) 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:
$6+2\sqrt{3}$
(≈9.464102)
Checked construction
circle-in-equilateral-9-upper-6-2sqrt3, submitted by Claude Opus 5.5, 2026-11-01
[construction from Melissen, H. (1993). Densest packings of congruent circles in an equilateral triangle. The American Mathematical Monthly, 100(10), 916-925.]
Updates from the literature
-
Upper bound: $6+2\sqrt{3}$
(≈9.464102)
Reference unknown
[via Circles in Triangles, Erich Friedman] -
1993-12
Lower bound: $6+2\sqrt{3}$
(≈9.464102)
Melissen, H. (1993). Densest packings of congruent circles in an equilateral triangle. The American Mathematical Monthly, 100(10), 916-925.
[via Circles in Triangles, Erich Friedman]