Pentagons in Triangles

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

1.
           2.
           3.
s = 2.71618+
Trivial.
 
  		Letting t=√(5+√5),
s = √(705+48t√30–120t√6–99√5)/3t = 4.01077+
Found by Ethan Riddle and
Proved by Nicolas Campailla in May 2026. 		s = 4.12157+
Trivial.
 
 

4.
           5.
           6.
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 = 6.81134+
Found by Jonathan Viquerat
in May 2026. 		s = 7.08971+
Found by Jonathan Viquerat
in May 2026.

10.
           11.
           12.
s = 7.17240+
Found by Jonathan Viquerat
in May 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 = 8.58923+
Found by Jonathan Viquerat
in May 2026. 		s = 8.69300+
Found by Jonathan Viquerat
in May 2026.

16.
           17.
           18.
s = 9.29702+
Found by Jonathan Viquerat
in May 2026. 		s = 9.57846+
Found by Jonathan Viquerat
in May 2026. 		s = 9.78480+
Found by Jonathan Viquerat
in May 2026.

19.
           20.
           21.
s = 9.93813+
Found by Jonathan Viquerat
in May 2026. 		s = 10.10536+
Found by Jonathan Viquerat
in May 2026. 		s = 10.18892+
Found by Jonathan Viquerat
in May 2026.

22.
           23.
           24.
s = 10.75939+
Found by Jonathan Viquerat
in May 2026. 		s = 11.00262+
Found by Jonathan Viquerat
in May 2026. 		s = 11.18879+
Found by Jonathan Viquerat
in May 2026.

25.
           26.
s = 11.33243+
Found by Ethan Riddle
in May 2026. 		s = 11.47102+
Found by Jonathan Viquerat
in May 2026.