Pentagons in Circles

The following pictures show n unit pentagons packed inside the smallest known circle (with radius r).

1.
           2.
           3.
r = √((5+√5)/10) = .85065+
Trivial. 		r = √(5+2√5)/2 = 1.53884+
Trivial. 		r = (1+√5)/2 = 1.61803+
Trivial.

4.
           5.
           6.
r = 1.82971+
Found by Jonathan Viquerat
in April 2026. 		r = 2.03271+
Found by Jonathan Viquerat
in April 2026. 		r = √((25+11√5)/10) = 2.22703+
Found by Jonathan Viquerat
in April 2026.

7.
           8.
           9.
r = 2.29470+
Found by Jake Loyd
in June 2026. 		r = 2.48128+
Found by Jonathan Viquerat
in April 2026. 		r = 2.70169+
Found by Jake Loyd
in June 2026.

10.
           11.
           12.
r = 2.79768+
Found by Jonathan Viquerat
in April 2026. 		r = 2.90171+
Found by Jake Loyd
in June 2026. 		r = 3.01808+
Found by Jake Loyd
in June 2026.

13.
           14.
           15.
r = 3.07768+
Found by Jake Loyd
in June 2026. 		r = 3.23377+
Found by Jonathan Viquerat
in June 2026. 		r = 3.33877+
Found by Jake Loyd
in June 2026.

16.
           17.
           18.
r = 3.43657+
Found by Jonathan Viquerat
in April 2026. 		r = 3.50888+
Found by Jake Loyd
in June 2026. 		r = 3.61619+
Found by Jake Loyd
in June 2026.

19.
           20.
           21.
r = 3.67189+
Found by Jake Loyd
in June 2026. 		r = 3.80052+
Found by Jake Loyd
in June 2026. 		r = 3.90693+
Found by Haowei Lin
in July 2026.

22.
           23.
           24.
r = 3.99781+
Found by Haowei Lin
in July 2026. 		r = 4.07193+
Found by Haowei Lin
in July 2026. 		r = 4.12402+
Found by Haowei Lin
in July 2026.

25.
           26.
           27.
r = 4.23686+
Found by Joshua Wolk
in September 2026. 		r = 4.29463+
Found by Joost de Winter
in September 2026. 		r = 4.35567+
Found by Joost de Winter
in September 2026.

28.
           29.
           30.
r = 4.46120+
Found by Joost de Winter
in September 2026. 		r = 4.52357+
Found by Joost de Winter
in September 2026. 		r = 4.59837+
Found by Joost de Winter
in September 2026.

31.
           32.
           33.
r = 4.68785+
Found by Joost de Winter
in September 2026. 		r = 4.74404+
Found by Joost de Winter
in September 2026. 		r = 4.80575+
Found by Joost de Winter
in September 2026.

34.
           35.
           36.
r = 4.86925+
Found by Joost de Winter
in September 2026. 		r = 4.92490+
Found by Joost de Winter
in September 2026. 		r = 4.99366+
Found by Joost de Winter
in September 2026.

37.
           38.
           39.
r = 5.06524+
Found by Joost de Winter
in September 2026. 		r = 5.14528+
Found by Joost de Winter
in September 2026. 		r = 5.21093+
Found by Joost de Winter
in September 2026.

40.
           41.
           42.
r = 5.28976+
Found by Joost de Winter
in September 2026. 		r = 5.33820+
Found by Joost de Winter
in September 2026. 		r = 5.39228+
Found by Joost de Winter
in September 2026.

43.
           44.
           45.
r = 5.45727+
Found by Joost de Winter
in September 2026. 		r = 5.51432+
Found by Joost de Winter
in September 2026. 		r = 5.56383+
Found by Joost de Winter
in September 2026.

46.
           47.
           48.
r = 5.62425+
Found by Joost de Winter
in September 2026. 		r = 5.68435+
Found by Joost de Winter
in September 2026. 		r = 5.73691+
Found by Joost de Winter
in September 2026.

49.
           50.
r = 5.79675+
Found by Joost de Winter
in September 2026. 		r = 5.86758+
Found by Joost de Winter
in September 2026.