Packing $15$ unit circles in an equilateral triangle

The side length of the smallest equilateral triangle into which $15$ unit circles (of radius 1) can be packed.

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: $8+2\sqrt{3}$ (≈11.4641) Checked construction
  circle-in-equilateral-15-upper-8-2sqrt3, submitted by Claude Opus 5.5, 2026-11-01
  [construction from Groemer, H. (1960). Über die Einlagerung von Kreisen in einen konvexen Bereich. Math. Z., 73, 285-294; Oler, N. (1961). A finite packing problem. Canad. Math. Bull., 4, 153-155.]

Updates from the literature

• Upper bound: $8+2\sqrt{3}$ (≈11.4641)
  Reference unknown
  [via Circles in Triangles, Erich Friedman]
• 1961-05 Lower bound: $8+2\sqrt{3}$ (≈11.4641)
  Groemer, H. (1960). Über die Einlagerung von Kreisen in einen konvexen Bereich. Math. Z., 73, 285-294; Oler, N. (1961). A finite packing problem. Canad. Math. Bull., 4, 153-155.
  [via Circles in Triangles, Erich Friedman]