Triangles in Squares

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

1.
           2.
           3.
s = (√2+√6)/4 = .965+
Trivial. 		s = √6/2 = 1.224+
Trivial. 		s = √3/2+√6/4 = 1.478+
Found by Erich Friedman
in 1996.

4.
           5.
           6.
s = 1+1/√3 = 1.577+
Found by Erich Friedman
in 1996. 		s = 1.803+
Found by Erich Friedman
in 1996. 		s = 9/2–3√3/2 = 1.901+
Found by Erich Friedman
in 1996.

7.
           8.
           9.
s = 2
Found by Erich Friedman
in 1996. 		s = 3√3/2–1/2 = 2.098+
Found by Erich Friedman
in 1996. 		s = 2.287+
Found by Maurizio Morandi
in July 2008.

10.
           11.
           12.
s = 2.377+
Found by David W. Cantrell
in July 2002. 		s = 2.490+
Found by Maurizio Morandi
in June 2008. 		s = 2.558+
Found by Dominik Kamp
in April 2026.

13.
           14.
           15.
s = 2.595+
Found by David W. Cantrell
in July 2002. 		s = 2.726+
Found by David W. Cantrell
in July 2002. 		s = 2.8299+
Found by Ian Watson
in April 2026.

16.
           17.
           18.
s = 2.9000+
Found by Ian Watson
in April 2026. 		s = 2.982+
Found by Maurizio Morandi
in July 2008. 		s = 3.051+
Found by Maurizio Morandi
in July 2008.

19.
           20.
           21.
s = 3.12930+
Found by Ian Watson
in April 2026. 		s = 3.2305+
Found by Ian Watson
in April 2026. 		s = 3.3111+
Found by Ian Watson
in April 2026.

22.

s = 3.3757+
Found by Ian Watson
in April 2026. 		           23.

s = 3.4311+
Found by Ian Watson
in April 2026. 		           24.

s = 3.468+
Found by Maurizio Morandi
in August 2008.

25.

s = 3.537+
Found by David W. Cantrell
in August 2012. 		           26.

s = 3.575+
Found by Maurizio Morandi
in September 2008. 		           27.

s = 3.6726+
Found by Ian Watson
in April 2026.

28.

s = 3.7571+
Found by Ian Watson
in April 2026. 		           29.

s = 3.818+
Found by Ignacio Vallejo
in April 2026. 		           30.

s = 3.8662+
Found by Ian Watson
in April 2026.

31.

s = 3.936+
Found by Emerson Connelly
in April 2026. 		           32.

s = 6–7√3/6 = 3.979+
Found by Maurizio Morandi
in May 2026. 		           33.

s = 4.058+
Found by Emerson Connelly
in April 2026.

34.

s = 4.098+
Found by Emerson Connelly
in April 2026. 		           35.

s = 4.14260+
Found by Jake Loyd
in May 2026. 		           36.

s = 4.214+
Found by Emerson Connelly
in April 2026.

37.

s = 4.27233+
Found by Nathan Roll
in April 2026. 		           38.

s = 4.32228+
Found by Jake Loyd
in May 2026. 		           39.

s = 4.37628+
Found by Jake Loyd
in May 2026.

40.

s = 4.43086+
Found by Jake Loyd
in May 2026. 		           41.

s = 4.469348+
Found by Emerson Connelly
in May 2026. 		           42.

s = 4.529257+
Found by Emerson Connelly
in May 2026.

43.

s = 4.581099+
Found by Emerson Connelly
in May 2026. 		           44.

s = 4.664740+
Found by Emerson Connelly
in May 2026. 		           45.

s = 4.715056+
Found by Emerson Connelly
in May 2026.