Triangles Covering Circles

The following pictures show n unit equilateral triangles covering the largest known circle (of radius r).


1.            2.            3.
r = √3/6 = .288+
Trivial.
r = √3/4 = .433+
Found by Erich Friedman
in 1997.
r = (38√3 - 9) / 109 = .521+
Found by Maurizio Morandi
in June 2009.


4.            5.            6.
r = (9√3 - 2√22) / 10 = .620+
Found by Trevor Green
in July 1999.
r = (79√3 - 3√79) / 152 = .724+
Found by Maurizio Morandi
in June 2009.
r = √3/2 = .866+
Trivial.


7.            8.            9.
r = .892+
Found by Maurizio Morandi
in July 2009.
r = (453√3 - 3√347) / 752 = .969+
Found by Maurizio Morandi
in June 2009.
r = (81√3 - 6√30) / 106 = 1.013+
Found by David Cantrell
in July 2005.


10.            11.            12.
r = (31√3 - √103) / 40 = 1.088+
Found by David Cantrell
in July 2005.
r = (177√3 - 2√586) / 226 = 1.142+
Found by David Cantrell
in July 2005.
r = (7√3 - √7) / 8 = 1.184+
Found by David Cantrell
in July 2005.


13.            14.            15.
r = (40√3 - √165) / 45 = 1.254+
Found by Maurizio Morandi
in August 2009.
r = 1.290+
Found by Maurizio Morandi
in August 2009.
r = 1.343+
Found by Maurizio Morandi
in August 2009.


16.            17.            18.
r = 1.392+
Found by Maurizio Morandi
in August 2009.
r = 1.434+
Found by Maurizio Morandi
in August 2009.
r = 1.475+
Found by Maurizio Morandi
in August 2009.


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