Squares in Pentagons

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

1.
           2.
           3.
s = cos(27°) / cos(18°)
s = .936+
Trivial.
  		s = 4√(1–2/√5)+√5–2
s = 1.535+
Found by Erich Friedman
in March 2012. 		s = 1.695+
Found by David W. Cantrell
in July 2012.
 

4.
           5.
           6.
s = 1.861+
Found by Maurizio Morandi
in July 2012.
  		s = 1 + cos(18°)
s = 1.951+
Found by Erich Friedman
in March 2012. 		s = 6√(1–2/√5)+√5–2
s = 2.185+
Found by Erich Friedman
in March 2012.

7.
           8.
           9.
s = 6√(1–2/√5)+2√5–4
s = 2.421+
Found by Erich Friedman
in July 2012. 		s = 2.565+
Found by David W. Cantrell
in July 2012.
  		s = 2.693+
Found by Timo Berthold et al
in January 2026.
 

10.
           11.
           12.
s = 2.793+
Found by Timo Berthold et al
in January 2026. 		s = 2.890+
Found by Timo Berthold et al
in January 2026. 		s = 2.969+
Found by Timo Berthold et al
in January 2026.

13.
           14.
           15.
s = 3.12074+
Found by Jake Loyd
in June 2026.
  		s = 3.187+
Found by David W. Cantrell
in July 2012.
  		s = (√(125–38√5)+8–2√5)/3
s = 3.284+
Found by David W. Cantrell
in July 2012.

16.
           17.
           18.
s = 3.47851+
Found by Jonathan Viquerat
in May 2026. 		s = 3.541+
Found by Ignacio Vallejo
in March 2026. 		s = 3.626+
Found by Ignacio Vallejo
in March 2026.

19.
           20.
           21.
s = 3.70918+
Found by Jake Loyd
in June 2026. 		s = 3.79775+
Found by Jake Loyd
in June 2026. 		s = 3.86365+
Found by Jake Loyd
in June 2026.

22.
           23.
           24.
s = 3.93347+
Found by Jake Loyd
in June 2026. 		s = 4.00109+
Found by Jake Loyd
in June 2026. 		s = 4.10103+
Found by Jake Loyd
in June 2026.

25.
           26.
           27.
s = 4.16727+
Found by Jake Loyd
in June 2026. 		s = 4.22150+
Found by Jake Loyd
in June 2026. 		s = 4.34851+
Found by Jake Loyd
in June 2026.

28.
           29.
           30.
s = 4.41576+
Found by Jake Loyd
in June 2026. 		s = 4.49879+
Found by Jake Loyd
in June 2026. 		s = 4.54753+
Found by Jake Loyd
in June 2026.

31.
           32.
           33.
s = 4.62404+
Found by Jake Loyd
in June 2026. 		s = 4.70000+
Found by Jake Loyd
in June 2026. 		s = 4.75331+
Found by Jake Loyd
in June 2026.

34.
           35.
           36.
s = 4.81345+
Found by Jake Loyd
in June 2026. 		s = 4.86283+
Found by Jake Loyd
in June 2026. 		s = 4.93131+
Found by Jake Loyd
in June 2026.

37.
           38.
           39.
s = 4.97801+
Found by Jake Loyd
in June 2026. 		s = 5.04838+
Found by Jake Loyd
in June 2026. 		s = 5.13653+
Found by Jake Loyd
in June 2026.

40.
s = 5.16849+
Found by Jake Loyd
in June 2026.