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 = 2.166+
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 = 2.32289+
Found by Ian Watson
in April 2026. 		s = 7/3 = 2.333+
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 = 2.400
Found by Emerson Connelly
in April 2026. 		s = 2.465+
Found by Emerson Connelly
in April 2026.

33.
           34.-35.
           36.
s = 2.498+
Found by Emerson Connelly
in April 2026. 		s = 5/2 = 2.500
Found by Emerson Connelly
in April 2026. 		s = 2.59123+
Found by Nathan Roll
in April 2026.

37.
           38.-39.
           40.
s = 2.61943+
Found by Nathan Roll
in April 2026. 		s = 8/3 = 2.666+
Found by Emerson Connelly
in April 2026. 		s = 2.71622+
Found by Nathan Roll
in April 2026.

41.
           42.
           43.
s = 2.74470+
Found by Emerson Connelly
in May 2026. 		s = 2.78403+
Found by Emerson Connelly
in May 2026. 		s = 2.82986+
Found by Emerson Connelly
in May 2026.

44.-45.
           46.
           47.
s = 17/6 = 2.833+
Found by Emerson Connelly
in May 2026. 		s = 29/10 = 2.900
Found by Emerson Connelly
in May 2026. 		s = 35/12 = 2.917+
Found by Emerson Connelly
in May 2026.

48.
           49.
           50.-54.
s = 2.94944+
Found by Emerson Connelly
in May 2026. 		s = 2.99203+
Found by Emerson Connelly
in May 2026. 		s = 3
Trivial.