Pentagons in Circles

The following pictures show n regular pentagons packed inside a unit circle, so that the total sum of the sides of the pentagons is as large as possible.

1.
           2.
           3.
Σ s = √((5–√5)/2) = 1.17557+
Trivial.
  		Σ s = 4√(1–2/√5) = 1.29967+
Trivial.
A continuum of solutions. 		Σ s = 1.85411+
Found by Jonathan Viquerat
in June 2026.

4.
           5.
           6.
Σ s = 2.23179+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 2.51913+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 2.79197+
Found by Jonathan Viquerat
in June 2026.

7.
           8.
           9.
Σ s = 3.06697+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 3.31241+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 3.51482+
Found by Jonathan Viquerat
in June 2026.

10.
           11.
           12.
Σ s = 3.71141+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 3.91715+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 4.11652+
Found by Jonathan Viquerat
in June 2026.

13.
           14.
           15.
Σ s = 4.29822+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 4.49438+
Found by Haowei Lin
in June 2026. 		Σ s = 4.65507+
Found by Haowei Lin
in June 2026.

16.
           17.
           18.
Σ s = 4.84434+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 5.00531+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 5.16108+
Found by Haowei Lin
in June 2026.

19.
           20.
           21.
Σ s = 5.31132+
Found by Haowei Lin
in June 2026. 		Σ s = 5.45458+
Found by Jonathan Viquerat
in September 2026. 		Σ s = 5.58046+
Found by Haowei Lin
in June 2026.

22.
           23.
           24.
Σ s = 5.71970+
Found by Pedro Martins
in October 2026. 		Σ s = 5.84892+
Found by Pedro Martins
in October 2026. 		Σ s = 5.98787+
Found by Pedro Martins
in October 2026.

25.
           26.
           27.
Σ s = 6.11442+
Found by Pedro Martins
in October 2026. 		Σ s = 6.25211+
Found by Pedro Martins
in October 2026. 		Σ s = 6.37074+
Found by Pedro Martins
in October 2026.

28.
           29.
           30.
Σ s = 6.49412+
Found by Pedro Martins
in October 2026. 		Σ s = 6.61255+
Found by Pedro Martins
in October 2026. 		Σ s = 6.73489+
Found by Joost de Winter
in October 2026.

31.
           32.
           33.
Σ s = 6.85434+
Found by Pedro Martins
in October 2026. 		Σ s = 6.97000+
Found by Pedro Martins
in October 2026. 		Σ s = 7.07888+
Found by Pedro Martins
in October 2026.

34.
           35.
           36.
Σ s = 7.19218+
Found by Pedro Martins
in October 2026. 		Σ s = 7.30204+
Found by Pedro Martins
in October 2026. 		Σ s = 7.41882+
Found by Pedro Martins
in October 2026.

37.
           38.
           39.
Σ s = 7.53322+
Found by Pedro Martins
in October 2026. 		Σ s = 7.64183+
Found by Pedro Martins
in October 2026. 		Σ s = 7.74295+
Found by Pedro Martins
in October 2026.

40.
           41.
           42.
Σ s = 7.84649+
Found by Pedro Martins
in October 2026. 		Σ s = 7.95318+
Found by Pedro Martins
in October 2026. 		Σ s = 8.05021+
Found by Pedro Martins
in October 2026.

43.
           44.
           45.
Σ s = 8.13861+
Found by Pedro Martins
in October 2026. 		Σ s = 8.23731+
Found by Pedro Martins
in October 2026. 		Σ s = 8.33136+
Found by Pedro Martins
in October 2026.

46.
           47.
           48.
Σ s = 8.42817+
Found by Joost de Winter
in October 2026. 		Σ s = 8.52562+
Found by Pedro Martins
in October 2026. 		Σ s = 8.62030+
Found by Pedro Martins
in October 2026.

49.
           50.
Σ s = 8.71188+
Found by Pedro Martins
in October 2026. 		Σ s = 8.81047+
Found by Pedro Martins
in October 2026.