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 = 1.385+
Found by David W. Cantrell
in July 2002.
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 = 1.609+
Found by DC after MM
in August 2012.
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 = √13/2 = 1.802+
Found by Erich Friedman
in 1997.


19.
           20.
           21.
r = 1.849+
Found by Maurizio Morandi
in June 2008.
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 = 1.988+
Found by Maurizio Morandi
in June 2008.
r = 2
Trivial.


25.
           26.
           27.
r = 2.055+
Found by David W. Cantrell
in July 2005.
r = 2.109+
Found by Maurizio Morandi
in April 2008.
r = 2.138+
Found by David W. Cantrell
in July 2005.


28.
           29.
           30.
r = 2.191+
Found by David W. Cantrell
in July 2005.
r = 2.215+
Found by Maurizio Morandi
in June 2008.
r = 2.258+
Found by Maurizio Morandi
in July 2008.


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


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