Triangles in Pentagons

The following pictures show n unit squares packed inside the smallest known pentagon (with side s).

1.
           2.
           3.
s = cos(42°) / cos(18°)
s = .781+
Trivial.
  		s = √(6/(3+√5))
s = 1.070+
Trivial.
  		s = 1.143+
Found by Maurizio Morandi
in August 2012.
 

4.
           5.
           6.
s = 1.199+
Found by Maurizio Morandi
in July 2012.
  		s = 1 + 2sin(12°)/√3
s = 1.240+
Found by Erich Friedman
in March 2012. 		s = 1.393+
Found by Dominik Kamp
in April 2026.

7.
           8.
           9.
s = 1.581+
Found by David W. Cantrell
in July 2012. 		s = 1.659+
Found by Maurizio Morandi
in July 2012. 		s = 1.767+
Found by Maurizio Morandi in July 2012.
Improved slightly by David W. Cantrell in August 2012.

10.
           11.
           12.
s = 1.815+
Found by Timo Berthold et al
in January 2026. 		s = 1.871+
Found by Timo Berthold et al
in January 2026. 		s = 1.973+
Found by Ignacio Vallejo
in March 2026.

13.
           14.
           15.
s = 2.017+
Found by Timo Berthold et al
in January 2026. 		s = 2.088+
Found by Ignacio Vallejo
in April 2026. 		s = 2.123+
Found by Ignacio Vallejo
in March 2026.

16.
           17.
           18.
s = 2.17598+
Found by Ian Watson
in April 2026. 		s = 2.23658+
Found by Ian Watson
in April 2026. 		s = 2.29382+
Found by Ian Watson
in April 2026.

19.
           20.
           21.
s = 2.344+
Found by Ignacio Vallejo
in March 2026. 		s = 2.43350+
Found by Ian Watson
in April 2026. 		s = 2.479+
Found by Ignacio Vallejo
in March 2026.

22.
           23.
           24.
s = 2.56740+
Found by Ian Watson
in April 2026. 		s = 2.61831+
Found by Ian Watson
in April 2026. 		s = 2.66641+
Found by Ian Watson
in April 2026.

25.
           26.
           27.
s = 2.720+
Found by Ignacio Vallejo
in March 2026. 		s = 2.771+
Found by Ignacio Vallejo
in March 2026. 		s = 2.81761+
Found by Ian Watson
in April 2026.

28.
           29.
           30.
s = 2.868+
Found by Ignacio Vallejo
in March 2026. 		s = 2.903+
Found by Ignacio Vallejo
in April 2026. 		s = 2.952+
Found by Ignacio Vallejo
in April 2026.

31.
           32.
           33.
s = 2.992+
Found by Emerson Connelly
in April 2026. 		s = 3.029+
Found by Emerson Connelly
in April 2026. 		s = 3.078+
Found by Emerson Connelly
in April 2026.

34.
           35.
           36.
s = 3.114+
Found by Emerson Connelly
in April 2026. 		s = 3.150+
Found by Emerson Connelly
in April 2026. 		s = 3.177+
Found by Emerson Connelly
in April 2026.

37.
           38.
           39.
s = 3.21529+
Found by Nathan Roll
in April 2026. 		s = 3.25877+
Found by Nathan Roll
in April 2026. 		s = 3.295+
Found by Emerson Connelly
in April 2026.

40.
s = 3.32786+
Found by Nathan Roll
in April 2026.