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 = 1.38468+
Found by Jonathan Viquerat
in June 2026. 		r = 1.466+
Found by David W. Cantrell
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 = 1.60448+
Found by Jake Loyd
in June 2026. 		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 = 1.80176+
Found by Jake Loyd
in June 2026.

19.
           20.
           21.
r = √(10/3) = 1.825+
Found by Haym Hirsh
in August 2019. 		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 = 1.98467+
Found by Jake Loyd
in June 2026. 		r = 2
Trivial.

25.
           26.
           27.
r = 2.055+
Found by David W. Cantrell
in July 2005. 		r = 2.10664+
Found by Thomas Schadt
in August 2026. 		r = 2.138+
Found by David W. Cantrell
in July 2005.

28.
           29.
           30.
r = 2.18805+
Found by Thomas Schadt
in August 2026. 		r = 2.20935+
Found by Jake Loyd
in June 2026. 		r = 2.25124+
Found by Thomas Schadt
in August 2026.

31.
           32.
           33.
r = 2.272+
Found by Maurizio Morandi
in July 2008. 		r = 2.31295+
Found by Thomas Schadt
in August 2026. 		r = 2.34169+
Found by Thomas Schadt
in August 2026.

34.
           35.
           36.
r = 2.37504+
Found by Thomas Schadt
in August 2026. 		r = 2.39906+
Found by Thomas Schadt
in August 2026. 		r = 2.43969+
Found by Thomas Schadt
in August 2026.

37.
           38.
           39.
r = 2.46349+
Found by Thomas Schadt
in August 2026. 		r = 2.49003+
Found by Thomas Schadt
in August 2026. 		r = 2.51938+
Found by Thomas Schadt
in August 2026.

40.
           41.
           42.
r = 2.55675+
Found by Thomas Schadt
in August 2026. 		r = 2.58629+
Found by Thomas Schadt
in August 2026. 		r = 2.61145+
Found by Thomas Schadt
in August 2026.

43.
           44.
           45.
r = 2.63998+
Found by Thomas Schadt
in August 2026. 		r = 2.66620+
Found by Thomas Schadt
in August 2026. 		r = 2.69809+
Found by Thomas Schadt
in August 2026.

46.
           47.
           48.
r = 2.72457+
Found by Thomas Schadt
in August 2026. 		r = 2.75037+
Found by Thomas Schadt
in August 2026. 		r = 2.77672+
Found by Thomas Schadt
in August 2026.

49.
           50.
r = 2.80521+
Found by Thomas Schadt
in August 2026. 		r = 2.82736+
Found by Thomas Schadt
in August 2026.