Pentagons in Triangles

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

1.
           2.
           3.
s = (1+√5)/2 + √((5+√5)/6)
s = 2.71621+
Trivial.
 
√((40+8√5)/3) – (3–√5)/2
s = 4.01077+
Found by Ethan Riddle and
Proved by Nicolas Campailla in May 2026. 		s = √((75+33√5)/8) – (3–√5)/4
s = 4.12164+
Found by Jonathan Viquerat
in May 2026.

4.
           5.
           6.
s = √(365+151√5)/2√6–3/4+√5/4
s = 5.21983+
Found by Jonathan Viquerat
in May 2026. 		s = 5.55809+
Found by Jonathan Viquerat
in May 2026. 		s = 5.63540+
Found by Jonathan Viquerat
in May 2026.

7.
           8.
           9.
s = 6.50034+
Found by Jonathan Viquerat
in May 2026. 		s = 6.81133+
Found by Bhavithran Ananthan
in July 2026. 		s = 7.08969+
Found by Bhavithran Ananthan
in July 2026.

10.
           11.
           12.
s = 7.17235+
Found by Bhavithran Ananthan
in July 2026. 		s = 7.93748+
Found by Jonathan Viquerat
in May 2026. 		s = 8.19117+
Found by Jonathan Viquerat
in May 2026.

13.
           14.
           15.
s = 8.37807+
Found by Jonathan Viquerat
in May 2026. 		s = 8.58917+
Found by Haowei Lin
in July 2026. 		s = 8.69161+
Found by Jake Loyd
in June 2026.

16.
           17.
           18.
s = 9.29699+
Found by Beppe Björkland
in October 2026. 		s = 9.57797+
Found by Jake Loyd
in June 2026. 		s = 9.78077+
Found by Jake Loyd
in June 2026.

19.
           20.
           21.
s = 9.92567+
Found by Haowei Lin
in July 2026. 		s = 10.09879+
Found by Jake Loyd
in June 2026. 		s = 10.17148+
Found by Bhavithran Ananthan
in July 2026.

22.
           23.
           24.
s = 10.75817+
Found by Beppe Björkland
in October 2026. 		s = 10.99097+
Found by Yanis Gomes
in October 2026. 		s = 11.17511+
Found by Jake Loyd
in June 2026.

25.
           26.
           27.
s = 11.32392+
Found by Jake Loyd
in June 2026. 		s = 11.45496+
Found by Joshua Wolk
in September 2026. 		s = 11.59777+
Found by Joshua Wolk
in September 2026.

28.
           29.
           30.
s = 11.69138+
Found by Beppe Björkland
in October 2026. 		s = 12.17916+
Found by Beppe Björkland
in October 2026. 		s = 12.38240+
Found by Yue Huang
in August 2026.

31.
           32.
           33.
s = 12.57719+
Found by Joshua Wolk
in September 2026. 		s = 12.70488+
Found by Joshua Wolk in September 2026. 		s = 12.85817+
Found by Joshua Wolk
in September 2026.

34.
           35.
           36.
s = 12.97324+
Found by Joshua Wolk
in September 2026. 		s = 13.09887+
Found by Joshua Wolk
in September 2026. 		s = 13.17841+
Found by Yue Huang
in August 2026.

37.
           38.
           39.
s = 13.62475+
Found by Yanis Gomes
in October 2026. 		s = 13.81858+
Found by Joost de Winter in September 2026. 		s = 13.97521+
Found by Yue Huang
in August 2026.

40.
           41.
           42.
s = 14.12492+
Found by Joshua Wolk
in September 2026. 		s = 14.23798+
Found by Joshua Wolk in September 2026. 		s = 14.37320+
Found by Joshua Wolk in September 2026.

43.
           44.
           45.
s = 14.47898+
Found by Joshua Wolk
in September 2026. 		s = 14.59237+
Found by Joshua Wolk in September 2026. 		s = 14.66451+
Found by Joshua Wolk in September 2026.

46.
           47.
           48.
s = 15.05751+
Found by Yanis Gomes
in October 2026. 		s = 15.23840+
Found by Beppe Björkland
in October 2026. 		s = 15.40024+
Found by Joshua Wolk in September 2026.

49.
           50.
s = 15.51379+
Found by Joshua Wolk in September 2026. 		s = 15.66032+
Found by Joshua Wolk in September 2026.