Squares in Circles

The following pictures show n unit squares packed inside the smallest known circle (of radius r). Only the cases n=1-7 have been proved optimal.

1.
           2.
           3.
r = 1 / √2 = .70710+
Trivial. 		r = √5 / 2 = 1.11803+
Trivial. 		r = 5 √17 / 16 = 1.28847+
Proved by The-Anh Vu-Le
in September 2026.

4.
           5.
           6.
r = √2 = 1.41421+
Proved by the International
Math Summer Camp in 2026. 		r = √10 / 2 = 1.58113+
Proved by The-Anh Vu-Le
in September 2026. 		r = 1.688+*
Proved by The-Anh Vu-Le
in September 2026.
* Exact value of √(19706163 + 13275064√2 – 40√(443374242065 + 313512226176√2) ) / 534 found by David Ellsworth in 2023.

7.
           8.
           9.
r = √13 / 2 = 1.80277+
Proved by The-Anh Vu-Le
in September 2026. 		r = 1.97877+
Found by David W. Cantrell
in March 2002. 		r = √1105 / 16 = 2.07759+
Found by Erich Friedman
in 1997.

10.
           11.
           12.
r = 3 / √2 = 2.12132+
Found by Erich Friedman
in 1997. 		r = 2.21386+
Found by Milan Kovacic
in May 2006. 		r = √5 = 2.23606+
Found by Erich Friedman
in 1997.

13.
           14.
           15.
r = 2.3607+
Found by David W. Cantrell
in November 2010. 		r = 5/2 = 2.5
Found by Erich Friedman
in 1997. 		r = 2.533+
Found by David W. Cantrell
in June 2002.

16.
           17.
           18.
r = √11009 / 40 = 2.62309+
Found by Erich Friedman
in 1997. 		r = 2.67687+
Found by Milan Kovacic
in May 2026. 		r = √481 / 8 = 2.74146+
Found by David W. Cantrell
in March 2002.

19.
           20.
           21.
r = 2.80150+
Found by Milan Kovacic
in May 2026. 		r = 2.893+
Found by David W. Cantrell
in June 2002. 		r = √34 / 2 = 2.91547+
Found by Erich Friedman
in 1997.

22.
           23.
           24.
r = 3.03239+
Found by Haowei Lin
in July 2026. 		r = 3.06738+
Found by Milan Kovacic
in May 2026. 		r = 3.10940+
Found by Milan Kovacic
in May 2026.

25.
           26.
           27.
r = 3.17450+
Found by Haowei Lin
in July 2026. 		r = √41 / 2 = 3.20156+
Found by Erich Friedman
in 1997. 		r = 3.26050+
Found by Milan Kovacic
in May 2026.

28.
           29.
           30.
r = 3.33147+
Found by Milan Kovacic
in May 2026. 		r = 3.39309+
Found by Milan Kovacic
in May 2026. 		r = 3.45962+
Found by Milan Kovacic
in May 2026.

31.
           32.
           33.
r = 3.52279+
Found by Daniel Lu
in September 2026. 		r = 3.57340+
Found by Milan Kovacic
in May 2026. 		r = 3.60010+
Found by Haowei Lin
in July 2026.

34.
           35.
           36.
r = 3.62852+
Found by Daniel Lu
in September 2026. 		r = √53 / 2 = 3.64005+
Found by David Eppstein
in March 2002. 		r = 3.72853+
Found by Joshua Wolk
in September 2026.

37.
           38.
           39.
r = 3.77126+
Found by Ephraim Duncan
in September 2026. 		r = 3.83514+
Found by Beppe Björkland
in September 2026. 		r = 3.88461+
Found by Beppe Björkland
in September 2026.

40.
           41.
           42.
r = 3.93833+
Found by Ephraim Duncan
in September 2026. 		r = 3.98879+
Found by Beppe Björkland
in October 2026. 		r = √65/2 = 4.03112+
Found by Yanis Gomes
in October 2026.

43.
           44.
           45.
r = 4.06733+
Found by Joshua Wolk
in September 2026. 		r = 4.09999+
Found by Joshua Wolk
in September 2026. 		r = 4.14067+
Found by Beppe Björkland
in October 2026.

46.
           47.
           48.
r = 4.16573+
Found by Joshua Wolk
in September 2026. 		r = 4.19870+
Found by Joshua Wolk
in September 2026. 		r = 4.25610+
Found by Joshua Wolk
in September 2026.

49.
           50.
r = 4.28307+
Found by Joshua Wolk
in September 2026. 		r = 4.30116+
Found by Joshua Wolk
in September 2026.