Squares in Circles

The following pictures show n squares packed inside a unit circle, so that the total sum of the sides of the squares is as large as possible.

1.            2.            3.
Σ s = √2 = 1.414+
Trivial. 		Σ s = 4 / √5 = 1.788+
Trivial. 		Σ s = 2.331+
Found by Erich Friedman
in June 2011.

4.            5.            6.
Σ s = 2√2 = 2.828+
Found by Erich Friedman
in June 2011. 		Σ s = 3.204+
Found by Maurizio Morandi
in July 2011. 		Σ s = 3.597+
Found by Maurizio Morandi
in July 2011.

7.            8.            9.
Σ s = (2√10 + √190) / 5 = 4.021+
Found by Erich Friedman
in June 2011. 		Σ s = (10√2 + 5√10 + √190) / 10 = 4.373+
Found by Erich Friedman
in June 2011. 		Σ s = (10√2 + 3√10) / 5 = 4.725+
Found by Erich Friedman
in June 2011.

10.            11.            12.
Σ s = 4.970+
Found by Erich Friedman
in June 2011. 		Σ s = 5.185+
Found by Maurizio Morandi
in July 2011. 		Σ s = 5.44255+
Found by Jonathan Viquerat
in June 2026.

13.            14.            15.
Σ s = 5.720+
Found by Maurizio Morandi
in August 2011. 		Σ s = 6.005+
Found by Erich Friedman
in June 2011. 		Σ s = 6.218+
Found by Maurizio Morandi
in July 2011.

16.            17.            18.
Σ s = 6.435+
Found by Maurizio Morandi
in July 2011. 		Σ s = 6.61435+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 6.82745+
Found by Jonathan Viquerat
in June 2026.

19.            20.            21.
Σ s = 7.069+
Found by David W. Cantrell
in July 2011. 		Σ s = 7.271+
Found by Maurizio Morandi
in July 2011. 		Σ s = 7.473+
Found by Maurizio Morandi
in July 2011.

22.            23.            24.
Σ s = 7.66770+
Found by Haowei Lin
in July 2026. 		Σ s = 7.86233+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 8.03235+
Found by Haowei Lin
in July 2026.

25.            26.            27.
Σ s = 8.20424+
Found by Haowei Lin
in July 2026. 		Σ s = 8.37795+
Found by Jonathan Viquerat
in October 2026. 		Σ s = 8.53680+
Found by Pedro Martins
in October 2026.

28.            29.            30.
Σ s = 8.70496+
Found by Jonathan Viquerat
in October 2026. 		Σ s = 8.88274+
Found by Joost de Winter
in September 2026. 		Σ s = 9.05382+
Found by Jonathan Viquerat
in October 2026.

31.            32.            33.
Σ s = 9.20941+
Found by Jonathan Viquerat
in October 2026. 		Σ s = 9.36414+
Found by Joost de Winter
in October 2026. 		Σ s = 9.51000+
Found by Jonathan Viquerat
in October 2026.

34.            35.            36.
Σ s = 9.65625+
Found by Pedro Martins
in October 2026. 		Σ s = 9.80548+
Found by Joost de Winter
in October 2026. 		Σ s = 9.95627+
Found by Joost de Winter
in October 2026.

37.            38.            39.
Σ s = 10.10867+
Found by Joost de Winter
in October 2026. 		Σ s = 10.25796+
Found by Joost de Winter
in October 2026. 		Σ s = 10.40462+
Found by Joost de Winter
in October 2026.

40.            41.            42.
Σ s = 10.55117+
Found by Joost de Winter
in October 2026. 		Σ s = 10.70337+
Found by Joost de Winter
in October 2026. 		Σ s = 10.84206+
Found by Joost de Winter
in October 2026.

43.            44.            45.
Σ s = 10.98076+
Found by Pedro Martins
in October 2026. 		Σ s = 11.11428+
Found by Pedro Martins
in October 2026. 		Σ s = 11.24625+
Found by Pedro Martins
in October 2026.

46.            47.            48.
Σ s = 11.37833+
Found by Pedro Martins
in October 2026. 		Σ s = 11.49632+
Found by Pedro Martins
in October 2026. 		Σ s = 11.61753+
Found by Joost de Winter
in October 2026.

49.            50.
Σ s = 11.74251+
Found by Pedro Martins
in October 2026. 		Σ s = 11.87228+
Found by Pedro Martins
in October 2026.