Triangles in Circles

The following pictures show n equilateral triangles with side 1 packed inside the smallest known circle (of radius r).


1.
           2.
           3.
r = 1/√3 = .577+
Trivial.
r = √3/2 = .866+
Trivial.
r = 3√3 – 3√2 = .953+
Found by Erich Friedman
in 1997.


4.
           5.
           6.
r = .980+
Found by Erich Friedman
in 1997.
r = 1
Trivial.
r = 1
Trivial.


7.
           8.
           9.
r = 1.152+
Found by Erich Friedman
in 1997.
r = 1.263+
Found by Maurizio Morandi
in April 2008.
r = 1.314+
Found by David W. Cantrell
in July 2002.


10.
           11.
           12.
r = 1.38468+
Found by Jonathan Viquerat
in June 2026.
r = 1.466+
Found by DC after MM
in August 2012.
r = 1.507+
Found by David W. Cantrell
in October 2005.


13.
           14.
           15.
r = √(7/3) = 1.527+
Found by Erich Friedman
in 1997.
r = 1.60448+
Found by Jake Loyd
in June 2026.
r = 1.636+
Found by David W. Cantrell
in July 2002.


16.
           17.
           18.
r = 1.687+
Found by Maurizio Morandi
in April 2008.
r = 1.730+
Found by David W. Cantrell
in July 2005.
r = 1.80176+
Found by Jake Loyd
in June 2026.


19.
           20.
           21.
r = √(10/3) = 1.825+
Found by Haym Hirsh
in August 2019.
r = 1.874+
Found by Erich Friedman
in 1997.
r = 1.910+
Found by Maurizio Morandi
in May 2008.


22.
           23.
           24.
r = 1.943+
Found by Maurizio Morandi
in June 2008.
r = 1.98467+
Found by Jake Loyd
in June 2026.
r = 2
Trivial.


25.
           26.
           27.
r = 2.055+
Found by David W. Cantrell
in July 2005.
r = 2.10665+
Found by Jake Loyd
in June 2026.
r = 2.138+
Found by David W. Cantrell
in July 2005.


28.
           29.
           30.
r = 2.18806+
Found by Jake Loyd
in June 2026.
r = 2.20935+
Found by Jake Loyd
in June 2026.
r = 2.25127+
Found by Jake Loyd
in June 2026.


31.
r = 2.272+
Found by Maurizio Morandi
in July 2008.