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 = 8.29236+
Found by Joost de Winter
in September 2026. 		Σ s = 8.50491+
Found by Joost de Winter
in September 2026.

28.            29.            30.
Σ s = 8.70495+
Found by Joost de Winter
in September 2026. 		Σ s = 8.88274+
Found by Joost de Winter
in September 2026. 		Σ s = 9.03014+
Found by Joost de Winter
in October 2026.

31.            32.            33.
Σ s = 9.12639+
Found by Joost de Winter
in October 2026. 		Σ s = 9.23231+
Found by Joost de Winter
in September 2026. 		Σ s = 9.38011+
Found by Joost de Winter
in October 2026.

34.            35.            36.
Σ s = 9.54226+
Found by Joost de Winter
in September 2026. 		Σ s = 9.77176+
Found by Joost de Winter
in September 2026. 		Σ s = 9.94451+
Found by Joost de Winter
in September 2026.

37.            38.            39.
Σ s = 10.08231+
Found by Joost de Winter
in October 2026. 		Σ s = 10.21885+
Found by Joost de Winter
in October 2026. 		Σ s = 10.36917+
Found by Joost de Winter
in September 2026.

40.            41.            42.
Σ s = 10.5383+
Found by Joost de Winter
in September 2026. 		Σ s = 10.67146+
Found by Joost de Winter
in October 2026. 		Σ s = 10.79793+
Found by Joost de Winter
in September 2026.

43.            44.            45.
Σ s = 10.95455+
Found by Joost de Winter
in September 2026. 		Σ s = 11.08416+
Found by Joost de Winter
in October 2026. 		Σ s = 11.17540+
Found by Joost de Winter
in October 2026.

46.            47.            48.
Σ s = 11.29949+
Found by Joost de Winter
in September 2026. 		Σ s = 11.43912+
Found by Joost de Winter
in October 2026. 		Σ s = 11.53085+
Found by Joost de Winter
in October 2026.

49.            50.
Σ s = 11.60861+
Found by Joost de Winter
in October 2026. 		Σ s = 11.71658+
Found by Joost de Winter
in October 2026.