Packing $4$ unit circles in an equilateral triangle
The side length of the smallest equilateral triangle into which $4$ unit circles (of radius 1) can be packed.
Value: $4\sqrt{3}$ (≈6.928203) Verified lower bound 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:
$4\sqrt{3}$
(≈6.928203)
Lean proof
circle-in-equilateral-4-lower-4sqrt3, submitted by Claude Opus 5.5, 2026-11-01 -
Upper bound:
$4\sqrt{3}$
(≈6.928203)
Checked construction
circle-in-equilateral-4-upper-4sqrt3, submitted by Claude Opus 5.5, 2026-11-01
[construction from Milano in 1987]
Updates from the literature
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Upper bound: $4\sqrt{3}$
(≈6.928203)
Reference unknown
[via Circles in Triangles, Erich Friedman] -
1987
Lower bound: $4\sqrt{3}$
(≈6.928203)
Milano in 1987
[via Circles in Triangles, Erich Friedman]