Circles in Pentagons

The following pictures show n circles packed inside a unit pentagon, so that the total sum of the radii of the circles is as large as possible.

1.
           2.
           3.
Σ r = cot(π/5)/2 = .68819+
Trivial.
  		Σ r = .76085+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 1.03457+
Found by Jonathan Viquerat
in June 2026.

4.
           5.
           6.
Σ r = 1.25099+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 5/(2tan(π/5)+2) = 1.44798+
Found by Jonathan Viquerat
in June 2026. 		Σ r = (4+csc(π/5))/(2tan(π/5)+2) = 1.65107+
Found by Jonathan Viquerat
in June 2026.

7.
           8.
           9.
Σ r = 1.74099+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 1.85250+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 1.95668+
Found by Jonathan Viquerat
in June 2026.

10.
           11.
           12.
Σ r = 2.06920+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 2.20261+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 2.29901+
Found by Jonathan Viquerat
in June 2026.

13.
           14.
           15.
Σ r = 2.40969+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 2.51439+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 2.61271+
Found by Joost de Winter
in September 2026.

16.
           17.
           18.
Σ r = 2.72429+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 2.78356+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 2.85730+
Found by Jonathan Viquerat
in June 2026.

19.
           20.
           21.
Σ r = 2.93757+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 3.01869+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 3.08715+
Found by Richard Wang
in October 2026.

22.
           23.
           24.
Σ r = 3.16566+
Found by Aaden Abraham
in October 2026. 		Σ r = 3.24734+
Found by Richard Wang
in October 2026. 		Σ r = 3.32758+
Found by Jonathan Viquerat
in October 2026.

25.
           26.
           27.
Σ r = 3.39164+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 3.47663+
Found by Richard Wang
in October 2026. 		Σ r = 3.53987+
Found by Jonathan Viquerat
in October 2026.

28.
           29.
           30.
Σ r = 3.61027+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 3.68134+
Found by Richard Wang
in October 2026. 		Σ r = 3.74588+
Found by Jonathan Viquerat
in October 2026.

31.
           32.
           33.
Σ r = 3.81681+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 3.86196+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 3.91731+
Found by Jonathan Viquerat
in October 2026.

34.
           35.
           36.
Σ r = 3.97647+
Found by Richard Wang
in October 2026. 		Σ r = 4.03575+
Found by Joost de Winter
in October 2026. 		Σ r = 4.09469+
Found by Jonathan Viquerat
in October 2026.

37.
           38.
           39.
Σ r = 4.14946+
Found by Joost de Winter
in October 2026. 		Σ r = 4.21143+
Found by Richard Wang
in October 2026. 		Σ r = 4.26611+
Found by Jonathan Viquerat
in October 2026.

40.
           41.
           42.
Σ r = 4.32634+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 4.38705+
Found by Aaden Abraham
in October 2026. 		Σ r = 4.44142+
Found by Jonathan Viquerat
in October 2026.

43.
           44.
           45.
Σ r = 4.49633+
Found by Richard Wang
in October 2026. 		Σ r = 4.55427+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 4.60620+
Found by Joost de Winter
in October 2026.

46.
           47.
           48.
Σ r = 4.66314+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 4.71466+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 4.76452+
Found by Jonathan Viquerat
in October 2026.

49.
           50.
Σ r = 4.81492+
Found by Joost de Winter
in October 2026. 		Σ r = 4.86154+
Found by Jonathan Viquerat
in October 2026.