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 = 1.385+

Found by David W. Cantrell

in July 2002.
r = 1.466+

Found by DC after MM

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 = 1.609+

Found by DC after MM

in August 2012.
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 = √13/2 = 1.802+

Found by Erich Friedman

in 1997.

19.

20.

21.

r = 1.849+

Found by Maurizio Morandi

in June 2008.
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 = 1.988+

Found by Maurizio Morandi

in June 2008.
r = 2

Trivial.

25.

26.

27.

r = 2.055+

Found by David W. Cantrell

in July 2005.
r = 2.109+

Found by Maurizio Morandi

in April 2008.
r = 2.138+

Found by David W. Cantrell

in July 2005.

28.

29.

30.

r = 2.191+

Found by David W. Cantrell

in July 2005.
r = 2.215+

Found by Maurizio Morandi

in June 2008.
r = 2.258+

Found by Maurizio Morandi

in July 2008.

31.

r = 2.272+

Found by Maurizio Morandi

in July 2008.