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.

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.