Triangles in Tans

The following pictures show n unit equilateral triangles packed inside the smallest known tan (with short side length s).

1.

s = √(3/2) = 1.22474+
Trivial. 		           2.

s = √3 = 1.73205+
Trivial. 		           3.

s = (√3+1)/√2 = 1.93185+
Trivial.

4.

s = (3+√3)/2 = 2.36602+
Found by Jonathan Viquerat
in May 2026. 		           5.

s = (3√2+√6+2√3)/4 = 2.53905+
Found by Jonathan Viquerat
in May 2026. 		           6.

s = √2+√(3/2) = 2.63895+
Found by Jonathan Viquerat
in May 2026.
7.

s = 2.78972+
Found by Jonathan Viquerat
in May 2026. 		           8.

s = 3.07313+
Found by Jonathan Viquerat
in May 2026. 		           9.

s = 3.18252+
Found by Jonathan Viquerat
in May 2026.
10.

s = 3.34054+
Found by Joshua Wolk
in September 2026. 		           11.

s = 3.46951+
Found by Jonathan Viquerat
in May 2026. 		           12.

s = 3.62482+
Found by Joshua Wolk
in September 2026.
13.

s = 3.71162+
Found by Yanis Gomes
in October 2026. 		           14.

s = 3.88268+
Found by Jonathan Viquerat
in May 2026. 		           15.

s = 3.98382+
Found by Thomas Greenleaf
in June 2026.
16.

s = 5/4+3√2–√(177+90√2)/12 = 4.03900+
Found by Yanis Gomes
in October 2026. 		           17.

s = 4.20065+
Found by Aapo Lipponen
in October 2026. 		           18.

s = 4.27119+
Found by Aapo Lipponen
in October 2026.
19.

s = 4.45699+
Found by Joshua Wolk
in September 2026. 		           20.

s = 4.56123+
Found by Yanis Gomes
in October 2026. 		           21.

s = 4.64137+
Found by Joshua Wolk
in September 2026.
22.

s = 4.73161+
Found by Joshua Wolk
in September 2026. 		           23.

s = 4.82957+
Found by Yanis Gomes
in October 2026. 		           24.

s = 4.88409+
Found by Jonathan Viquerat
in September 2026.
25.

s = 5.01708+
Found by Joshua Wolk
in September 2026. 		           26.

s = 5.09949+
Found by Joshua Wolk
in September 2026. 		           27.

s = 5.19744+
Found by Joshua Wolk
in September 2026.
28.

s = 1/8+23√2/8+√(363+252√2)/24 = 5.30841+
Found by Yanis Gomes
in October 2026. 		           29.

s = 5.36431+
Found by Joshua Wolk
in September 2026. 		           30.

s = 5.44418+
Found by Joshua Wolk
in September 2026.
31.

s = 5.49091+
Found by Joshua Wolk
in September 2026. 		           32.

s = 5.60591+
Found by Joshua Wolk
in September 2026. 		           33.

s = 5.69919+
Found by Joshua Wolk
in September 2026.
34.

s = 5.75394+
Found by Joshua Wolk
in September 2026. 		           35.

s = 5.86274+
Found by Joshua Wolk
in September 2026. 		           36.

s = 5.94643+
Found by Joshua Wolk
in September 2026.
37.

s = 6.02387+
Found by Joshua Wolk
in September 2026. 		           38.

s = 6.09773+
Found by Joshua Wolk
in September 2026. 		           39.

s = 6.13509+
Found by Joshua Wolk
in September 2026.
40.

s = 6.23627+
Found by Joshua Wolk
in September 2026. 		           41.

s = 6.28514+
Found by Joshua Wolk
in September 2026. 		           42.

s = 6.33921+
Found by Yanis Gomes
in October 2026.
43.

s = 6.45222+
Found by Joshua Wolk
in September 2026. 		           44.

s = 6.52167+
Found by Yanis Gomes
in October 2026. 		           45.

s = 6.60810+
Found by Joshua Wolk
in September 2026.
46.

s = 6.65465+
Found by Yanis Gomes
in October 2026. 		           47.

s = 6.73663+
Found by Yanis Gomes
in October 2026. 		           48.

s = 6.78174+
Found by Joshua Wolk
in September 2026.
49.

s = 6.85189+
Found by Joshua Wolk
in September 2026. 		           50.

s = 6.91746+
Found by Joshua Wolk
in September 2026.