Triangles in Circles

The following pictures show n equilateral triangles with side 1 packed inside the smallest known circle (of radius r ). Most of these have not been proved optimal.

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 Cantrell
in July 2002.

10.            11.            12.
r = 1.385+
Found by David Cantrell
in July 2002. 		r = 1.467+
Found by Maurizio Morandi
in April 2008. 		r = 1.507+
Found by David Cantrell
in October 2005.

13.            14.            15.
r = √(7/3) = 1.527+
Found by Erich Friedman
in 1997. 		r = 1.610+
Found by Maurizio Morandi
in April 2008. 		r = 1.636+
Found by David Cantrell
in July 2002.

16.            17.            18.
r = 1.687+
Found by Maurizio Morandi
in April 2008. 		r = 1.730+
Found by David Cantrell
in July 2005. 		r = √13/2 = 1.802+
Found by Erich Friedman
in 1997.

19.            20.            21.
r = 1.862+
Found by Maurizio Morandi
in April 2008. 		r = 1.874+
Found by Erich Friedman
in 1997. 		r = 1.921+
Found by David Cantrell
in July 2005.

22.            23.            24.
r = 1.950+
Found by David Cantrell in
July 2005. 		r = 1.990+
Found by Maurizio Morandi
in April 2008. 		r = 2
Trivial.

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

28.            29.            30.
r = 2.191+
Found by David Cantrell
in July 2005. 		r = 2.252+
Found by Maurizio Morandi
in April 2008. 		r = 2.269+
Found by David Cantrell
in July 2005.

31.
r = 79√3/60 = 2.280+
Found by David Cantrell
in July 2005.

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