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 = (7√3 - 3√7) / 8 = .523+
Found by Erich Friedman
in 1997.

4. 5. 6.
r = (9√3 - 2√22) / 10 = .620+
Found by Trevor Green
in July 1999. r = 5√3/12 = .721+
Found by David Cantrell in
August 2002. r = √3/2 = .866+
Trivial.

7. 8. 9.
r = (15√3 - √21) / 24 = .891+
Found by David Cantrell in
July 2005. r = 19√3/34 = .967+
Found by David Cantrell in
July 2005. 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.

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1.		2.		3.
r = √3/6 = .288+ Trivial.		r = √3/4 = .433+ Found by Erich Friedman in 1997.		r = (7√3 - 3√7) / 8 = .523+ Found by Erich Friedman in 1997.

4.		5.		6.
r = (9√3 - 2√22) / 10 = .620+ Found by Trevor Green in July 1999.		r = 5√3/12 = .721+ Found by David Cantrell in August 2002.		r = √3/2 = .866+ Trivial.

7.		8.		9.
r = (15√3 - √21) / 24 = .891+ Found by David Cantrell in July 2005.		r = 19√3/34 = .967+ Found by David Cantrell in July 2005.		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.