Pentagons in Squares

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

1.
           2.
           3.
s = (3√2+√10+2√(5+√5))/8
s = 1.59811+
Trivial. 		s = (1+√5)[√(250+110√5) – 5(1+√5)]/8
s = 2.46345+
Trivial. 		s = (4+5√5/2)√(5+2√5)–55/4–23√5/4
s = 2.90811+
Found by Jonathan Viquerat
in April 2026.

4.
           5.
           6.
s = ((3–√5)√(5–√5)+7√10+√2)/8
s = 3.10252+
Found by Jonathan Viquerat
in April 2026. 		s = 3.54903+
Found by Jonathan Viquerat
in April 2026. 		s = 3.88299+
Found by Haowei Lin
in July 2026.

7.
           8.
           9.
s = 4.18037+
Found by Nicolas Campailla
in May 2026. 		s = 4.38190+
Found by Aristotle Papowitz
in July 2026. 		s = 4.60032+
Found by Yanis Gomes
in October 2026.

10.
           11.
           12.
s = 4.90581+
Found by Jonathan Viquerat
in June 2026. 		s = 5.11555+
Found by Jonathan Viquerat
in August 2026. 		s = 5.23221+
Found by Timo Berthold et al
in May 2026.

13.
           14.
           15.
s = 5.52102+
Found by Jake Loyd
in June 2026. 		s = 5.69659+
Found by Jake Loyd
in June 2026. 		s = 5.90118+
Found by Jonathan Viquerat
in August 2026.

16.
           17.
           18.
s = 6.06451+
Found by Aapo Lipponen
in October 2026. 		s = 6.23761+
Found by Jonathan Viquerat
in June 2026. 		s = 6.32565+
Found by Bhavithran Ananthan
in July 2026.

19.
           20.
           21.
s = 6.55781+
Found by Jake Loyd
in June 2026. 		s = 6.66135+
Found by Haowei Lin
in July 2026. 		s = 6.85368+
Found by Jake Loyd
in June 2026.

22.
           23.
           24.
s = 7.04445+
Found by Haowei Lin
in July 2026. 		s = 7.20624+
Found by Jake Loyd
in June 2026. 		s = 7.33690+
Found by Jake Loyd
in June 2026.

25.
           26.
           27.
s = 7.45359+
Found by Aapo Lipponen
in October 2026. 		s = 7.57345+
Found by Joshua Wolk
in September 2026. 		s = 7.66091+
Found by Joshua Wolk
in September 2026.

28.
           29.
           30.
s = 7.82006+
Found by Joshua Wolk
in September 2026. 		s = 7.98469+
Found by Joshua Wolk in September 2026. 		s = 8.08481+
Found by Joshua Wolk
in September 2026.

31.
           32.
           33.
s = 8.24522+
Found by Joshua Wolk
in September 2026. 		s = 8.37127+
Found by Joshua Wolk in September 2026. 		s = 8.53157+
Found by Joshua Wolk
in September 2026.

34.
           35.
           36.
s = 8.62755+
Found by Joshua Wolk
in September 2026. 		s = 8.67380+
Found by Joshua Wolk in September 2026. 		s = 8.83023+
Found by Yanis Gomes
in October 2026.

37.
           38.
           39.
s = 8.93171+
Found by Yanis Gomes
in October 2026. 		s = 9.01359+
Found by Joshua Wolk in September 2026. 		s = 9.07504+
Found by Joshua Wolk
in September 2026.

40.
           41.
           42.
s = 9.27375+
Found by Joshua Wolk
in September 2026. 		s = 9.40016+
Found by Beppe Björkland
in October 2026. 		s = 9.50319+
Found by Joshua Wolk
in September 2026.

43.
           44.
           45.
s = 9.64526+
Found by Yanis Gomes
in October 2026. 		s = 9.76066+
Found by Joshua Wolk in September 2026. 		s = 9.85102+
Found by Joshua Wolk
in September 2026.

46.
           47.
           48.
s = 9.92458+
Found by Joshua Wolk
in September 2026. 		s = 9.97327+
Found by Joshua Wolk in September 2026. 		s = 10.00817+
Found by Yanis Gomes
in October 2026.

49.
           50.
s = 10.20364+
Found by Joshua Wolk in September 2026. 		s = 10.31605+
Found by Yanis Gomes
in October 2026.