Pentagons in Squares

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

1.
           2.
           3.
Σ s = √(8–2√(10+2√5)) = .62573+
Trivial.
  		Σ s = .81187+
Trivial.
A continuum of solutions. 		Σ s = 1.08249+
Found by Jonathan Viquerat
in June 2026.

4.
           5.
           6.
Σ s = 1.33718+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 1.46940+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 1.61106+
Found by Jonathan Viquerat
in June 2026.

7.
           8.
           9.
Σ s = 1.75771+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 1.91200+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 2.04259+
Found by Jonathan Viquerat
in June 2026.

10.
           11.
           12.
Σ s = 2.15626+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 2.26772+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 2.38481+
Found by Jonathan Viquerat
in June 2026.

13.
           14.
           15.
Σ s = 2.48116+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 2.57744+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 2.67895+
Found by Pedro Martins
in October 2026.

16.
           17.
           18.
Σ s = 2.77356+
Found by Pedro Martins
in October 2026. 		Σ s = 2.85450+
Found by Pedro Martins
in October 2026. 		Σ s = 2.94083+
Found by Pedro Martins
in October 2026.

19.
           20.
           21.
Σ s = 3.03279+
Found by Pedro Martins
in October 2026. 		Σ s = 3.11739+
Found by Pedro Martins
in October 2026. 		Σ s = 3.20044+
Found by Pedro Martins
in October 2026.

22.
           23.
           24.
Σ s = 3.27490+
Found by Pedro Martins
in October 2026. 		Σ s = 3.35326+
Found by Pedro Martins
in October 2026. 		Σ s = 3.42351+
Found by Pedro Martins
in October 2026.

25.
           26.
           27.
Σ s = 3.4992+
Found by Pedro Martins
in October 2026. 		Σ s = 3.57451+
Found by Pedro Martins
in October 2026. 		Σ s = 3.64559+
Found by Pedro Martins
in October 2026.

28.
           29.
           30.
Σ s = 3.71910+
Found by Pedro Martins
in October 2026. 		Σ s = 3.78700+
Found by Pedro Martins
in October 2026. 		Σ s = 3.85471+
Found by Pedro Martins
in October 2026.

31.
           32.
           33.
Σ s = 3.91941+
Found by Pedro Martins
in October 2026. 		Σ s = 3.98005+
Found by Pedro Martins
in October 2026. 		Σ s = 4.04261+
Found by Pedro Martins
in October 2026.

34.
           35.
           36.
Σ s = 4.10332+
Found by Pedro Martins
in October 2026. 		Σ s = 4.16338+
Found by Pedro Martins
in October 2026. 		Σ s = 4.22269+
Found by Pedro Martins
in October 2026.

37.
           38.
           39.
Σ s = 4.28673+
Found by Pedro Martins
in October 2026. 		Σ s = 4.35197+
Found by Pedro Martins
in October 2026. 		Σ s = 4.41011+
Found by Pedro Martins
in October 2026.

40.
           41.
           42.
Σ s = 4.47313+
Found by Pedro Martins
in October 2026. 		Σ s = 4.52957+
Found by Pedro Martins
in October 2026. 		Σ s = 4.58484+
Found by Pedro Martins
in October 2026.

43.
           44.
           45.
Σ s = 4.63960+
Found by Pedro Martins
in October 2026. 		Σ s = 4.69342+
Found by Pedro Martins
in October 2026. 		Σ s = 4.74674+
Found by Pedro Martins
in October 2026.

46.
           47.
           48.
Σ s = 4.79944+
Found by Pedro Martins
in October 2026. 		Σ s = 4.85314+
Found by Pedro Martins
in October 2026. 		Σ s = 4.90243+
Found by Pedro Martins
in October 2026.

49.
           50.
Σ s = 4.95778+
Found by Pedro Martins
in October 2026. 		Σ s = 5.00517+
Found by Pedro Martins
in October 2026.