Circle packing in a circle

If more than one optimal solution exists, all are shown.^[1]

${\displaystyle n}$	Enclosing circle radius ${\displaystyle r}$	Density ${\displaystyle n\!/r^{2}}$	Optimality	Layout(s) of the ${\displaystyle n}$ circles
1	1	1.0000...	Trivially optimal.
2	2	0.5000...	Trivially optimal.
3	2.155...   ${\displaystyle 1+{\frac {2}{\sqrt {3}}}}$	0.6461...	Trivially optimal.
4	2.414...   ${\displaystyle 1+{\sqrt {2}}}$	0.6864...	Trivially optimal.
5	2.701...   ${\displaystyle 1+{\sqrt {2\left(1+{\frac {1}{\sqrt {5}}}\right)}}}$	0.6854...	Proved optimal by Graham (1968)[2]
6	3	0.6666...	Proved optimal by Graham (1968)[2]
7	3	0.7777...	Trivially optimal.
8	3.304...   ${\displaystyle 1+{\frac {1}{\sin {\frac {\pi }{7}}}}}$	0.7328...	Proved optimal by Pirl (1969)[3]
9	3.613...   ${\displaystyle 1+{\sqrt {2\left(2+{\sqrt {2}}\right)}}}$	0.6895...	Proved optimal by Pirl (1969)[3]
10	3.813...	0.6878...	Proved optimal by Pirl (1969)[3]
11	3.923...   ${\displaystyle 1+{\frac {1}{\sin {\frac {\pi }{9}}}}}$	0.7148...	Proved optimal by Melissen (1994)[4]
12	4.029...	0.7392...	Proved optimal by Fodor (2000)[5]
13	4.236...   ${\displaystyle 2+{\sqrt {5}}}$	0.7245...	Proved optimal by Fodor (2003)[6]
14	4.328...	0.7474...	Proved optimal by Ekanayake and LaFountain (2024).[7]
15	4.521...   ${\displaystyle 1\!+\!{\sqrt {6\!+\!{\frac {2}{\sqrt {5}}}\!+\!4{\sqrt {1\!+\!{\frac {2}{\sqrt {5}}}}}}}}$	0.7339...	Conjectured optimal by Pirl (1969).[8]
16	4.615...	0.7512...	Conjectured optimal by Goldberg (1971).[8]
17	4.792...	0.7403...	Conjectured optimal by Reis (1975).[8]
18	4.863...   ${\displaystyle 1+{\sqrt {2}}+{\sqrt {6}}}$	0.7609...	Conjectured optimal by Pirl (1969), with additional arrangements by Graham, Lubachevsky, Nurmela, and Östergård (1998).[8]
19	4.863...   ${\displaystyle 1+{\sqrt {2}}+{\sqrt {6}}}$	0.8032...	Proved optimal by Fodor (1999)[9]
20	5.122...	0.7623...	Conjectured optimal by Goldberg (1971).[8]

Only 26 optimal packings are thought to be rigid (with no circles able to "rattle"). Numbers in bold are prime:

• Proven for n = 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 19
• Conjectured for n = 15, 16, 17, 18, 22, 23, 27, 30, 31, 33, 37, 61, 91

Of these, solutions for n = 2, 3, 4, 7, 19, and 37 achieve a packing density greater than any smaller number > 1. (Higher density records all have rattles.)^[10]