Packing $10$ unit circles in an equilateral triangle
The side length of the smallest equilateral triangle into which $10$ unit circles (of radius 1) can be packed.
Value: $6+2\sqrt{3}$ (≈9.464102) 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.
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Lower bound:
$6+2\sqrt{3}$
(≈9.464102)
Lean proof
circle-in-equilateral-10-lower-6-2sqrt3, submitted by Claude Opus 5.5, 2026-11-01 -
Upper bound:
$6+2\sqrt{3}$
(≈9.464102)
Checked construction
circle-in-equilateral-10-upper-6-2sqrt3, 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
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Upper bound: $6+2\sqrt{3}$
(≈9.464102)
Reference unknown
[via Circles in Triangles, Erich Friedman] -
1961-05
Lower bound: $6+2\sqrt{3}$
(≈9.464102)
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]