Circles in Arbitrary Triangles

The following pictures show n unit circles packed inside the triangle with smallest known area A (with sides a, b, and c). For n not shown, the best known packing is in an equilateral triangle.

2.
           4.
           7.
A = 6 + 4√2 = 11.656+
a = 2 + 2√2     b = 2 + 2√2     c = 4 + 2√2
Found by David W. Cantrell in June 2007.
Proved by Tej Stead in June 2026. 		A = 9 + 6√3 = 19.392+
a = 3 + √3     b = 3 + 3√3     c = 6 + 2√3
Found by David W. Cantrell in June 2007.
  		A = 31.212+
a = 7.946+     b = 7.946+     c = 10.352+
Found by David W. Cantrell in June 2007.
 

8.
           11.
           12.
A = 35.936+
a = 8.038+     b = 9.018+     c = 11.270+
Found by David W. Cantrell in June 2007. 		A = 47.003+
a = 9.547+     b = 10.170+     c = 12.081+
Found by David W. Cantrell in June 2007. 		A = 51.304+
a = 9.875+     b = 10.666+     c = 12.792+
Found by David W. Cantrell in June 2007.

13.
           16.
           17.
A = 54.894+
a = 10.632+     b = 11.333+     c = 11.922+
Found by David W. Cantrell in June 2007. 		A = 21 + 26√3 = 66.033+
a = 7 + √3     b = 3 + 7√3     c = 14 + 2√3
Found by David W. Cantrell in June 2007. 		A = 70.275+
a = 11.673+     b = 12.434+     c = 14.775+
Found by David W. Cantrell in June 2007.

18.
           19.
           22.
A = 74.803+
a = 12.723+     b = 13.171+     c = 13.577+
Found by David W. Cantrell in June 2007. 		A = 77.346+
a = 12.892+     b = 13.169+     c = 14.130+
Found by David W. Cantrell in June 2007. 		A = 89.127+
a = 13.211+     b = 14.540+     c = 15.579+
Found by David W. Cantrell in June 2007.

23.
           25.
           26.
A = 92.74138+
a = 13.951+     b = 13.951+     c = 16.471+
Found by Tej Stead in June 2026. 		A = 27 + 42√3 = 99.74613+
a = 9 + √3     b = 3 + 9√3     c = 18 + 2√3
Found by Tej Stead in June 2026. 		A = 102.70487+
a = 15.154+     b = 15.408+     c = 15.651+
Found by Tej Stead in June 2026.

29.
           30.
           31.
A = 115.29224+
a = 15.252+     b = 16.465+     c = 17.456+
Found by Tej Stead in June 2026. 		A = 118.84862+
a = 15.202+     b = 16.810+     c = 18.054+
Found by Tej Stead in June 2026. 		A = 122.58804+
a = 15.153+     b = 17.191+     c = 18.682+
Found by Tej Stead in June 2026.

32.
           33.
           34.
A = 127.07490+
a = 16.149+     b = 17.249+     c = 18.172+
Found by Tej Stead in June 2026. 		A = 129.20635+
a = 16.725+     b = 17.309+     c = 17.840+
Found by Tej Stead in June 2026. 		A = 131.58592+
a = 17.289+     b = 17.289+     c = 17.728+
Found by Tej Stead in June 2026.

37.
           38.
           39.
A = 144.93603+
a = 14.943+     b = 19.755+     c = 22.398+
Found by Tej Stead in June 2026. 		A = 148.37141+
a = 17.243+     b = 18.696+     c = 19.871+
Found by Tej Stead in June 2026. 		A = 151.83073+
a = 17.205+     b = 18.992+     c = 20.379+
Found by Tej Stead in June 2026.

41.
           42.
           43.
A = 160.11055+
a = 18.664+     b = 19.263+     c = 19.810+
Found by Tej Stead in June 2026. 		A = 161.88709+
a = 18.730+     b = 19.374+     c = 19.960+
Found by Tej Stead in June 2026. 		A = 163.71933+
a = 19.349+     b = 19.445+     c = 19.539+
Found by Tej Stead in June 2026.

46.
           47.
           48.
A = 177.88737+
a = 19.307+     b = 20.364+     c = 21.277+
Found by Tej Stead in June 2026. 		A = 183.09423+
a = 18.894+     b = 20.856+     c = 22.379+
Found by Tej Stead in June 2026. 		A = 184.75052+
a = 19.243+     b = 20.862+     c = 22.171+
Found by Tej Stead in June 2026.

50.
A = 194.49896+
a = 20.783+     b = 21.209+     c = 21.612+
Found by Tej Stead in June 2026.