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 = 5.71416+
Found by Jonathan Viquerat
in September 2026. 		Σ s = 5.77735+
Found by Joost de Winter
in September 2026. 		Σ s = 5.95414+
Found by Joost de Winter
in September 2026.

25.
           26.
           27.
Σ s = 6.08091+
Found by Joost de Winter
in September 2026. 		Σ s = 6.17825+
Found by Joost de Winter
in October 2026. 		Σ s = 6.32048+
Found by Joost de Winter
in September 2026.

28.
           29.
           30.
Σ s = 6.43052+
Found by Joost de Winter
in September 2026. 		Σ s = 6.55172+
Found by Joost de Winter
in September 2026. 		Σ s = 6.67610+
Found by Joost de Winter
in September 2026.

31.
           32.
           33.
Σ s = 6.78602+
Found by Joost de Winter
in September 2026. 		Σ s = 6.89723+
Found by Joost de Winter
in September 2026. 		Σ s = 7.01183+
Found by Joost de Winter
in September 2026.

34.
           35.
           36.
Σ s = 7.15187+
Found by Joost de Winter
in October 2026. 		Σ s = 7.27105+
Found by Joost de Winter
in September 2026. 		Σ s = 7.40701+
Found by Joost de Winter
in September 2026.

37.
           38.
           39.
Σ s = 7.48164+
Found by Joost de Winter
in September 2026. 		Σ s = 7.58628+
Found by Joost de Winter
in September 2026. 		Σ s = 7.71147+
Found by Joost de Winter
in September 2026.

40.
           41.
           42.
Σ s = 7.78285+
Found by Joost de Winter
in October 2026. 		Σ s = 7.88005+
Found by Joost de Winter
in September 2026. 		Σ s = 7.95675+
Found by Joost de Winter
in October 2026.

43.
           44.
           45.
Σ s = 8.04710+
Found by Joost de Winter
in October 2026. 		Σ s = 8.18066+
Found by Joost de Winter
in October 2026. 		Σ s = 8.30681+
Found by Joost de Winter
in October 2026.

46.
           47.
           48.
Σ s = 8.42354+
Found by Joost de Winter
in October 2026. 		Σ s = 8.49232+
Found by Joost de Winter
in October 2026. 		Σ s = 8.55078+
Found by Joost de Winter
in October 2026.

49.
           50.
Σ s = 8.59910+
Found by Joost de Winter
in October 2026. 		Σ s = 8.65227+
Found by Joost de Winter
in October 2026.