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

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 = 6.81133+
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 = 9.29700+
Found by Haowei Lin
in July 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 = 10.75818+
Found by Jake Loyd
in June 2026. 		s = 10.99100+
Found by Jake Loyd
in June 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 = 11.45549+
Found by Aristotle Papowitz
in July 2026. 		s = 11.59953+
Found by Bhavithran Ananthan
in July 2026.

28.
           29.
           30.
s = 11.69416+
Found by Bhavithran Ananthan
in July 2026. 		s = 12.18243+
Found by Bhavithran Ananthan
in July 2026. 		s = 12.38893+
Found by Haowei Lin
in July 2026.

31.
           32.
           33.
s = 12.58515+
Found by Bhavithran Ananthan
in July 2026. 		s = 12.71117+
Found by Haowei Lin
in July 2026. 		s = 12.86391+
Found by Haowei Lin
in July 2026.

34.
           35.
           36.
s = 12.98141+
Found by Haowei Lin
in July 2026. 		s = 13.11409+
Found by Bhavithran Ananthan
in July 2026. 		s = 13.18354+
Found by Haowei Lin
in July 2026.

37.
           38.
           39.
s = 13.64359+
Found by Haowei Lin
in July 2026. 		s = 13.83966+
Found by Bhavithran Ananthan
in July 2026. 		s = 13.98431+
Found by Haowei Lin
in July 2026.

40.
s = 14.13953+
Found by Haowei Lin
in July 2026.