Squares in Circles

The following pictures show n unit squares packed inside the smallest known circle (of radius r). Only the cases n=1 and n=2 have been proved optimal.

1.
           2.
           3.
r = √2 / 2 = .707+
Trivial. 		r = √5 / 2 = 1.118+
Trivial. 		r = 5 √17 / 16 = 1.288+
Found by Erich Friedman
in 1997.

4.
           5.
           6.
r = √2 = 1.414+
Found by Erich Friedman
in 1997. 		r = √10 / 2 = 1.581+
Found by Erich Friedman
in 1997. 		r = 1.688+*
Found by Erich Friedman
in 1997.
* Exact value of √(19706163 + 13275064√2 – 40√(443374242065 + 313512226176√2) ) / 534 found by Lew Baxter in 2025.

7.
           8.
           9.
r = √13 / 2 = 1.802+
Found by Erich Friedman
in 1997. 		r = 1.978+
Found by David W. Cantrell
in March 2002. 		r = √1105 / 16 = 2.077+
Found by Erich Friedman
in 1997.

10.
           11.
           12.
r = 3 √2 / 2 = 2.121+
Found by Erich Friedman
in 1997. 		r = 2.214+
Found by David W. Cantrell
in March 2002. 		r = √5 = 2.236+
Found by Erich Friedman
in 1997.

13.
           14.
           15.
r = 2.3607+
Found by David W. Cantrell
in November 2010. 		r = 2.5
Found by Erich Friedman
in 1997. 		r = 2.533+
Found by by David W. Cantrell
in June 2002.

16.
           17.
           18.
r = √11009 / 40 = 2.623+
Found by Erich Friedman
in 1997. 		r = 2.677+
Found by Maurizio Morandi
in December 2010. 		r = √481 / 8 = 2.741+
Found by David W. Cantrell
in March 2002.

19.
           20.
           21.
r = 2.807+
Found by David W. Cantrell
in June 2002. 		r = 2.893+
Found by David W. Cantrell
in June 2002. 		r = √34 / 2 = 2.915+
Found by Erich Friedman
in 1997.

22.
           23.
           24.
r = 3.036+
Found by David W. Cantrell
in June 2002. 		r = 3.073+
Found by David W. Cantrell
in June 2002. 		r = 3.110+
Found by Maurizio Morandi
in October 2010.

25.
           26.
           27.
r = √5809 / 24 = 3.175+
Found by David W. Cantrell
in April 2002. 		r = √41 / 2 = 3.201+
Found by Erich Friedman
in 1997. 		r = 13√145/48 = 3.261+
Found by Maurizio Morandi
in September 2010.

28.
           29.
           30.
r = 3.3471+
Found by David W. Cantrell
in November 2010. 		r = 3.412+
Found by DC after MM
in August 2012. 		r = √27985 / 48 = 3.485+
Found by David W. Cantrell
in March 2002.

31.
           32.
           33.
r = 5 √2 / 2 = 3.535+
Found by Erich Friedman
in 1997. 		r = √40385 / 56 = 3.588+
Found by Erich Friedman
in 1997. 		r = √13 = 3.605+
Found by Maurizio Morandi
in June 2010.

34.
           35.
r = 3.629+
Found by Maurizio Morandi
in December 2010. 		r = √53 / 2 = 3.640+
Found by David Eppstein
in March 2002.