Squares in Octagons

The following pictures show n unit squares packed inside the smallest known octagon (with side s).

1.
           2.
           3.
s = 1/√(2+√2) = .54119+
Trivial.
  		s = √(6–√2–√(20√2–26))/2 = 0.87669+
Found by Maurizio Morandi
in December 2013. 		s = 18 – 12√2 = 1.02940+
Found by Erich Friedman
in December 2013.

4.
           5.
           6.
s = √(4–2√2) = 1.08239+
Trivial.
  		s = 3√2 – 3 = 1.24261+
Trivial.
  		s = 7 – 4√2 = 1.34314+
Found by Maurizio Morandi
in December 2013.

7.
           8.
           9.
s = √(5–√2–√(8√2–9)) = 1.43690+
Found by Maurizio Morandi
in December 2013. 		s = 1.56962+
Found by Grant Mowry
in May 2026. 		s = 3/√(2+√2) = 1.62358+
Trivial.
 

10.
           11.
           12.
s = 1.68171+
Found by Timo Berthold et al
in May 2026. 		s = 1.73152+
Found by Jonathan Viquerat
in May 2026. 		s = √(6–√2–√(20√2–26)) = 1.75339+
Found by Maurizio Morandi
in December 2013.

13.
           14.
           15.
s = 1.89674+
Found by Grant Mowry
in May 2026. 		s = 1.98302+
Found by Timo Berthold et al
in May 2026. 		s = 2.03168+
Found by Jonathan Viquerat
in May 2026.

16.-17.
           18.
           19.
s = 5√2 – 5 = 2.07106+
Trivial.
  		s = 2.16732+
Found by Timo Berthold et al
in May 2026. 		s = 2.22435+
Found by Jake Loyd
in May 2026.

20.
           21.
           22.
s = 2.28598+
Found by Jake Loyd
in May 2026. 		s = sqrt(48–9√2–3√(104√2–126))/2 = 2.31833+
Found by Jake Loyd
in May 2026. 		s = 2.41575+
Found by Jake Loyd
in May 2026.

23.
           24.
           25.
s = 2.46879+
Found by Jake Loyd
in May 2026. 		s = 6√2–6 = 2.48528+
Trivial. 		s = 17(2–√2)/4 = 2.48959+
Found by Jake Loyd
in May 2026.

26.
           27.
           28.
s = 2.52709+
Found by Aristotle Papowitz
in July 2026. 		s = 2.58584+
Found by Jake Loyd
in May 2026. 		s = 2.62788+
Found by Jake Loyd
in May 2026.

29.
           30.
           31.
s = 2.70553+
Found by Jake Loyd
in May 2026. 		s = 2(11–√2)/7 = 2.73879+
Found by Maurizio Morandi
in May 2026. 		s = 2.78428+
Found by Aristotle Papowitz
in July 2026.

32.
           33.
           34.
s = 2.83980+
Found by Aristotle Papowitz
in July 2026. 		s = 2.85903+
Found by Aristotle Papowitz
in July 2026. 		s = 7√2–7 = 2.89949+
Found by Mohamed Metwalli
in June 2026.

35.
           36.
           37.
s = 2.90679+
Found by Aristotle Papowitz
in July 2026. 		s = 2.92870+
Found by Aristotle Papowitz
in July 2026. 		s = 10–5√2 = 2.92893+
Trivial.
 

38.
           39.
           40.
s = 3.02113+
Found by Aristotle Papowitz
in July 2026. 		s = 3.08444+
Found by Aristotle Papowitz
in July 2026. 		s = 3.14201+
Found by Aristotle Papowitz
in July 2026.