Circle packing in a circle

Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle.

Minimum solutions (in case several minimal solutions have been shown to exist, only one variant appears in the table):[1]

Number of
unit circles
Enclosing circle radius Density Optimality Diagram
1 1 1.0000 Trivially optimal.
2 2 0.5000 Trivially optimal.
3 ${\displaystyle 1+{\frac {2}{\sqrt {3}}}}$ ≈ 2.154... 0.6466... Trivially optimal.
4 ${\displaystyle 1+{\sqrt {2}}}$ ≈ 2.414... 0.6864... Trivially optimal.
5 ${\displaystyle 1+{\sqrt {2\left(1+{\frac {1}{\sqrt {5}}}\right)}}}$ ≈ 2.701... 0.6854... Trivially optimal.
Also proved optimal by Graham
(1968)[2]
6 3 0.6667... Trivially optimal.
Also proved optimal by Graham
(1968)[2]
7 3 0.7778... Trivially optimal.
8 ${\displaystyle 1+{\frac {1}{\sin \left({\frac {\pi }{7}}\right)}}}$ ≈ 3.304... 0.7328... Proved optimal by Pirl
(1969)[3]
9 ${\displaystyle 1+{\sqrt {2\left(2+{\sqrt {2}}\right)}}}$ ≈ 3.613... 0.6895... Proved optimal by Pirl
(1969)[3]
10 3.813... 0.6878... Proved optimal by Pirl
(1969)[3]
11 ${\displaystyle 1+{\frac {1}{\sin \left({\frac {\pi }{9}}\right)}}}$ ≈ 3.923... 0.7148... Proved optimal by Melissen
(1994)[4]
12 4.029... 0.7392... Proved optimal by Fodor
(2000)[5]
13 ${\displaystyle 2+{\sqrt {5}}}$ ≈ 4.236... 0.7245... Proved optimal by Fodor
(2003)[6]
14 4.328... 0.7474... Conjectured optimal.[7]
15 ${\displaystyle 1+{\sqrt {6+{\frac {2}{\sqrt {5}}}+4{\sqrt {1+{\frac {2}{\sqrt {5}}}}}}}}$ ≈ 4.521... 0.7339... Conjectured optimal.[7]
16 4.615... 0.7512... Conjectured optimal.[7]
17 4.792... 0.7403... Conjectured optimal.[7]
18 ${\displaystyle 1+{\sqrt {2}}+{\sqrt {6}}}$ ≈ 4.863... 0.7611... Conjectured optimal.[7]
19 ${\displaystyle 1+{\sqrt {2}}+{\sqrt {6}}}$ ≈ 4.863... 0.8034... Proved optimal by Fodor
(1999)[8]
20 5.122... 0.7623... Conjectured optimal.[7]