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 = 3.34084+
Found by Jonathan Viquerat
in May 2026.
           11.

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

s = 3.62487+
Found by Bhavithran Ananthan
in July 2026.

13.

s = 3.71163+
Found by Jonathan Viquerat
in May 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 = 4.03901+
Found by Thomas Greenleaf
in June 2026.
           17.

s = 4.20127+
Found by Bhavithran Ananthan
in July 2026.
           18.

s = 4.27120+
Found by Thomas Greenleaf
in June 2026.

19.

s = 4.45701+
Found by Thomas Greenleaf
in June 2026.
           20.

s = 4.56124+
Found by Bhavithran Ananthan
in July 2026.
           21.

s = 4.64141+
Found by Akshaj Shandilya
in July 2026.

22.

s = 4.73276+
Found by Akshaj Shandilya
in July 2026.
           23.

s = 4.82974+
Found by Akshaj Shandilya
in July 2026.
           24.

s = 4.91168+
Found by Akshaj Shandilya
in July 2026.

25.

s = 5.05586+
Found by Akshaj Shandilya
in July 2026.
           26.

s = 5.10960+
Found by Akshaj Shandilya
in July 2026.
           27.

s = 5.19773+
Found by Akshaj Shandilya
in July 2026.

28.

s = 5.32367+
Found by Akshaj Shandilya
in July 2026.
           29.

s = 5.40552+
Found by Akshaj Shandilya
in July 2026.
           30.

s = 5.46262+
Found by Akshaj Shandilya
in July 2026.