Circles in Circles

The following pictures show n circles with the largest possible sum of radii packed inside a unit circle.

1.
           2.
           3.
Σ r = 1
Trivial. 		Σ r = 1
Trivial.
A continuum of solutions. 		Σ r = 6√3–9 = 1.392+
Trivial.

4.
           5.
           6.
Σ r = 5/3 = 1.666+
Found by Veit Elser
in November 2009. 		Σ r = 1.86684+
Found by Haowei Lin
in July 2026. 		Σ r = 1+3/(1+csc(π/5)) = 2.110+
Found by Veit Elser
in November 2009.

7.
           8.
           9.
Σ r = 7/3 = 2.333+
Found by Veit Elser
in November 2009. 		Σ r = 1+5/(1+csc(π/7)) = 2.512+
Found by Veit Elser
in November 2009. 		Σ r = 1+6/(1+csc(π/8)) = 2.660+
Found by Veit Elser
in November 2009.

10.
           11.
           12.
Σ r = 1+7/(1+csc(π/9)) = 2.783+
Found by David W. Cantrell
in April 2011. 		Σ r = 2.947+
Found by Veit Elser
in November 2009. 		Σ r = 3.103+
Found by Veit Elser
in November 2009.

13.
           14.
           15.
Σ r = 3.227+
Found by David W. Cantrell
in April 2011. 		Σ r = 3.359+
Found by Veit Elser
in November 2009. 		Σ r = 3.480+
Found by David W. Cantrell
in April 2011.

16.
           17.
           18.
Σ r = 3.611+
Found by David W. Cantrell
in April 2011. 		Σ r = 3.730+
Found by David W. Cantrell
in April 2011. 		Σ r = 3.850+
Found by David W. Cantrell
in April 2011.

19.
           20.
           21.
Σ r = 3.981+
Found by David W. Cantrell
in May 2011. 		Σ r = 4.076+
Found by David W. Cantrell
in April 2011. 		Σ r = 4.181+
Found by David W. Cantrell
in June 2011.

22.
           23.
           24.
Σ r = 4.287+
Found by David W. Cantrell
in June 2011. 		Σ r = 4.372+
Found by David W. Cantrell
in June 2011. 		Σ r = 4.466+
Found by David W. Cantrell
in June 2011.

25.
           26.
           27.
Σ r = 4.564+
Found by David W. Cantrell
in June 2011. 		Σ r = 4.664+
Found by David W. Cantrell
in June 2011. 		Σ r = 4.762+
Found by David W. Cantrell
in June 2011.

28.
           29.
           30.
Σ r = 4.849+
Found by David W. Cantrell
in June 2011. 		Σ r = 4.934+
Found by David W. Cantrell
in June 2011. 		Σ r = 5.019+
Found by David W. Cantrell
in June 2011.

31.
           32.
           33.
Σ r = 5.104+
Found by David W. Cantrell
in June 2011. 		Σ r = 5.191+
Found by David W. Cantrell
in June 2011. 		Σ r = 5.27748+
Found by Jonathan Viquerat
in September 2026.

34.
           35.
           36.
Σ r = 5.36044+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 5.44625+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 5.53033+
Found by Jonathan Viquerat
in September 2026.

37.
           38.
           39.
Σ r = 5.62172+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 5.69249+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 5.76337+
Found by Jonathan Viquerat
in September 2026.

40.
           41.
           42.
Σ r = 5.83631+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 5.91033+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 5.98166+
Found by Jonathan Viquerat
in September 2026.

43.
           44.
           45.
Σ r = 6.04710+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 6.11393+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 6.18800+
Found by Jonathan Viquerat
in September 2026.

46.
           47.
           48.
Σ r = 6.26015+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 6.32779+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 6.40862+
Found by Jonathan Viquerat
in September 2026.

49.
           50.
Σ r = 6.47331+
Found by Jonathan Viquerat
in September 2026. 		Σ r = 6.53486+
Found by Jonathan Viquerat
in September 2026.