Triangles in Hexagons

The following pictures show n equilateral triangles with side 1 packed inside the smallest known regular hexagon (of side length s ).

1.
           2.
           3.
s = 1/√3 = .577+
Trivial. 		s = √3/2 = .866+
Trivial. 		s = √((6+√33)/12) = .989+
Found by Erich Friedman
in April 2005.

4.
           5.
           6.
s = 1
Trivial. 		s = 1
Trivial. 		s = 1
Trivial.

7.
           8.
           9.
s = 1.277+
Found by David W. Cantrell
in May 2005. 		s = 1.356+
Found by David W. Cantrell
in July 2012. 		s = 1.434+
Found by Erich Friedman
in April 2005.

10.
           11.
           12.
s = 1.497+
Found by Timo Berthold et al
in January 2026. 		s = 3/2 = 1.500
Found by Erich Friedman
in April 2005. 		s = 1.623+
Found by Timo Berthold et al
in January 2026.

13.
           14.
           15.
s = √(587+51√17)/12√2 = 1.663829+
Found by Ian Watson
in April 2026. 		s = 1.725+
Found by David W. Cantrell
in August 2012. 		s = 7/4 = 1.750
Found by Erich Friedman
in April 2005.

16.
           17.
           18.
s = 9/5 = 1.800
Found by Maurizio Morandi
in April 2008. 		s = 11/6 = 1.833+
Found by Maurizio Morandi
in April 2008. 		s = 17/9 = 1.888+
Found by Maurizio Morandi
in April 2008.

19.
           20.
           21.
s = 1.953+
Found by Alex Adrián Nieto
in April 2026. 		s = 1.97591+
Found by Aaden Abraham
in April 2026. 		s = 1.99878+
Found by Ian Watson
in April 2026.

22.-24.
           25.
           26.
s = 2
Trivial.
  		s = 13/6 = 2.16666+
Found by Ignacio Vallejo
in April 2026. 		s = 2.21991+
Found by Ian Watson
in April 2026.

27.
           28.
           29.
s = 2.29541+
Found by Ian Watson
in April 2026. 		s = 2.32251+
Found by Jake Loyd
in June 2026. 		s = 7/3 = 2.33333+
Found by Ignacio Vallejo
in April 2026.

30.
           31.
           32.
s = 19/8 = 2.375
Found by Ignacio Vallejo
in April 2026. 		s = 12/5 = 2.4
Found by Emerson Connelly
in April 2026. 		s = 2.46121+
Found by Jake Loyd
in June 2026.

33.
           34.-35.
           36.
s = 2.49038+
Found by Jake Loyd
in June 2026. 		s = 5/2 = 2.5
Found by Emerson Connelly
in April 2026. 		s = 2.58309+
Found by Thomas Greenleaf
in June 2026.

37.
           38.
           39.
s = 2.61436+
Found by Jake Loyd
in June 2026. 		s = 2.65998+
Found by Jake Loyd
in June 2026. 		s = 8/3 = 2.66666+
Found by Emerson Connelly
in April 2026.

40.
           41.
           42.
s = 2.71131+
Found by Jake Loyd
in June 2026. 		s = 2.73692+
Found by Jake Loyd
in June 2026. 		s = 11/4 = 2.75
Found by Maurizio Morandi
in May 2026.

43.
           44.-45.
           46.
s = 14/5 =2.8
Found by Thomas Greenleaf
in June 2026. 		s = 17/6 = 2.833+
Found by Emerson Connelly
in May 2026. 		s = 2.88830+
Found by Jake Loyd
in June 2026.

47.
           48.
           49.
s = 35/12 = 2.917+
Found by Emerson Connelly
in May 2026. 		s = 47/16 =2.9375
Found by Thomas Greenleaf
in June 2026. 		s = 2.97693+
Found by Thomas Greenleaf
in June 2026.

50.-54.
s = 3
Trivial.