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Covering radii of the best known spherical coverings by $n$ equal caps
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Numbers
$n$
normalisation 
$\rho_n$ or $r_n$
1
$\rho_n$:
180
comment: One cap covers the sphere only when it includes its antipodal point.
1
$r_n$:
2
comment: One cap covers the sphere only when it includes its antipodal point.
2
$\rho_n$:
90
comment: Two antipodal cap centres attain the optimal covering radius.
2
$r_n$:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
comment: Two antipodal cap centres attain the optimal covering radius.
3
$\rho_n$:
90
comment: Two antipodal cap centres and any third distinct centre attain the optimal covering radius.
3
$r_n$:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
comment: Two antipodal cap centres and any third distinct centre attain the optimal covering radius.
4
$\rho_n$:
70.52877936550930863075400066003756403699315689909205171182889364396025356193960443663812871163941235
comment: Regular tetrahedron: $\cos(\pi\rho_4/180)=1/3$. Optimality is proven [1].
4
$r_n$:
1.154700538379251529018297561003914911295203502540253752037204652967955344605866691387430791171499050
comment: Regular tetrahedron: $\cos(\pi\rho_4/180)=1/3$. Optimality is proven [1].
5
$\rho_n$:
63.43494882292201064842780627954670532879578570035477897201398368471193417851525938829142618939783528
comment: Triangular bipyramid: $\cos(\pi\rho_5/180)=1/\sqrt5$. Optimality is proven [2].
5
$r_n$:
1.051462224238267212051338169695753214570995864486683563057871046482422292806428036743265257663105142
comment: Triangular bipyramid: $\cos(\pi\rho_5/180)=1/\sqrt5$. Optimality is proven [2].
6
$\rho_n$:
54.73561031724534568462299966998121798150342155045397414408555317801987321903019778168093564418029382
comment: Regular octahedron: $\cos(\pi\rho_6/180)=1/\sqrt3$. Optimality is proven [1].
6
$r_n$:
0.9194016867619661219552680198089906002488253271464188132155935141313730717288459801827508386430217513
comment: Regular octahedron: $\cos(\pi\rho_6/180)=1/\sqrt3$. Optimality is proven [1].
7
$\rho_n$:
51.02655266313435024347849134243076346621594114324817621752364865242110357678317249550907851277024025
comment: Pentagonal bipyramid: $\cos(\pi\rho_7/180)=\sqrt{(3+\sqrt5)/(11+\sqrt5)}$. Optimality is proven [2].
7
$r_n$:
0.8614404568568926255588500295973543053783960546758181672434336509504439838535859036914410338209460652
comment: Pentagonal bipyramid: $\cos(\pi\rho_7/180)=\sqrt{(3+\sqrt5)/(11+\sqrt5)}$. Optimality is proven [2].
8
$\rho_n$:
48.1395291
comment: Schütte's arrangement. Optimality is proven [3]; these entries retain numerical source precision.
8
$r_n$:
0.81569739
comment: Schütte's arrangement. Optimality is proven [3]; these entries retain numerical source precision.
9
$\rho_n$:
45.8788879
9
$r_n$:
0.77951605
10
$\rho_n$:
42.30782663010037070614892594603633033868065280854159410579871340474704040081461821995461399795241272
comment: Two poles and two staggered squares: $\cos(\pi\rho_{10}/180)=\sqrt{(1+\sqrt2)/(3+\sqrt2)}$. Optimality is proven [4].
10
$r_n$:
0.7217490502158837314819359610954276927713123020888220598724858536433705423566700500129581204843361896
comment: Two poles and two staggered squares: $\cos(\pi\rho_{10}/180)=\sqrt{(1+\sqrt2)/(3+\sqrt2)}$. Optimality is proven [4].
11
$\rho_n$:
41.4271960
11
$r_n$:
0.70739368
12
$\rho_n$:
37.37736814064969564285713849493751757009504738187193663730661502897439747871990088035727257050183788
comment: Regular icosahedron: $\cos(\pi\rho_{12}/180)=\sqrt{(5+2\sqrt5)/15}$. Optimality is proven [1].
12
$r_n$:
0.6408518201709875412850519828984058745745773522186724195983235171004484812664746491335597076084089104
comment: Regular icosahedron: $\cos(\pi\rho_{12}/180)=\sqrt{(5+2\sqrt5)/15}$. Optimality is proven [1].
13
$\rho_n$:
37.0685428
13
$r_n$:
0.63574368
14
$\rho_n$:
34.93792692312262013480504690000047906141610123413081656465812442222221822183380298395619328698072903
comment: Two poles and two staggered regular hexagons: $\cos(\pi\rho_{14}/180)=\sqrt{(3+3\sqrt3)/(7+3\sqrt3)}$. Optimality is proven [4].
14
$r_n$:
0.6003782733651972531993437387949391485324868575675556195055858578394122371917549792079478712994474452
comment: Two poles and two staggered regular hexagons: $\cos(\pi\rho_{14}/180)=\sqrt{(3+3\sqrt3)/(7+3\sqrt3)}$. Optimality is proven [4].
15
$\rho_n$:
34.0399002
15
$r_n$:
0.58540933
16
$\rho_n$:
32.8988128
16
$r_n$:
0.56633714
17
$\rho_n$:
32.0929329
17
$r_n$:
0.55283368
18
$\rho_n$:
31.0131719
18
$r_n$:
0.53469828
19
$\rho_n$:
30.3686749
19
$r_n$:
0.52385074
20
$\rho_n$:
29.6230959
20
$r_n$:
0.51128123
21
$\rho_n$:
28.8244769
21
$r_n$:
0.49779354
22
$\rho_n$:
27.8100588
22
$r_n$:
0.48062650
23
$\rho_n$:
27.4818687
23
$r_n$:
0.47506440
24
$\rho_n$:
26.8126364
24
$r_n$:
0.46371035
25
$\rho_n$:
26.3287855
25
$r_n$:
0.45549156
26
$\rho_n$:
25.8449223
26
$r_n$:
0.44726445
27
$\rho_n$:
25.2509549
27
$r_n$:
0.43715435
28
$\rho_n$:
24.6589490
28
$r_n$:
0.42706595
29
$\rho_n$:
24.3683987
29
$r_n$:
0.42211049
30
$\rho_n$:
23.8787580
30
$r_n$:
0.41375332
31
$\rho_n$:
23.6119921
31
$r_n$:
0.40919698
32
$\rho_n$:
22.6904805
32
$r_n$:
0.39344073
33
$\rho_n$:
22.5905117
33
$r_n$:
0.39172989
34
$\rho_n$:
22.3314637
34
$r_n$:
0.38729523
35
$\rho_n$:
22.0725569
35
$r_n$:
0.38286100
36
$\rho_n$:
21.6994390
36
$r_n$:
0.37646728
37
$\rho_n$:
21.3100299
37
$r_n$:
0.36979014
38
$\rho_n$:
21.0698585
38
$r_n$:
0.36566983
39
$\rho_n$:
20.8511245
39
$r_n$:
0.36191589
40
$\rho_n$:
20.4721354
40
$r_n$:
0.35540851
41
$\rho_n$:
20.3177153
41
$r_n$:
0.35275595
42
$\rho_n$:
20.0480918
42
$r_n$:
0.34812293
43
$\rho_n$:
19.8428334
43
$r_n$:
0.34459463
44
$\rho_n$:
19.6375705
44
$r_n$:
0.34106514
45
$\rho_n$:
19.4207407
45
$r_n$:
0.33733557
46
$\rho_n$:
19.1586113
46
$r_n$:
0.33282522
47
$\rho_n$:
18.9924594
47
$r_n$:
0.32996541
48
$\rho_n$:
18.6892566
48
$r_n$:
0.32474489
49
$\rho_n$:
18.5926797
49
$r_n$:
0.32308156
50
$\rho_n$:
18.3000226
50
$r_n$:
0.31803976
51
$\rho_n$:
18.1990012
51
$r_n$:
0.31629892
52
$\rho_n$:
18.0544758
52
$r_n$:
0.31380797
53
$\rho_n$:
17.8845735
53
$r_n$:
0.31087900
54
$\rho_n$:
17.6791448
54
$r_n$:
0.30733668
55
$\rho_n$:
17.5222393
55
$r_n$:
0.30463040
56
$\rho_n$:
17.3501140
56
$r_n$:
0.30166095
57
$\rho_n$:
17.1758476
57
$r_n$:
0.29865388
58
$\rho_n$:
17.0199611
58
$r_n$:
0.29596338
59
$\rho_n$:
16.9034031
59
$r_n$:
0.29395130
60
$\rho_n$:
16.7719330
60
$r_n$:
0.29168144
61
$\rho_n$:
16.6391846
61
$r_n$:
0.28938912
62
$\rho_n$:
16.4906597
62
$r_n$:
0.28682391
63
$\rho_n$:
16.3679364
63
$r_n$:
0.28470396
64
$\rho_n$:
16.1940191
64
$r_n$:
0.28169912
65
$\rho_n$:
16.1114062
65
$r_n$:
0.28027155
66
$\rho_n$:
15.9550615
66
$r_n$:
0.27756949
67
$\rho_n$:
15.8581808
67
$r_n$:
0.27589487
68
$\rho_n$:
15.7236959
68
$r_n$:
0.27356991
69
$\rho_n$:
15.5950401
69
$r_n$:
0.27134538
70
$\rho_n$:
15.4951288
70
$r_n$:
0.26961762
71
$\rho_n$:
15.3918905
71
$r_n$:
0.26783211
72
$\rho_n$:
15.1445321
72
$r_n$:
0.26355316
73
$\rho_n$:
15.1164438
73
$r_n$:
0.26306719
74
$\rho_n$:
15.0311866
74
$r_n$:
0.26159203
75
$\rho_n$:
14.9454278
75
$r_n$:
0.26010804
76
$\rho_n$:
14.8539208
76
$r_n$:
0.25852442
77
$\rho_n$:
14.7449905
77
$r_n$:
0.25663906
78
$\rho_n$:
14.6550578
78
$r_n$:
0.25508234
79
$\rho_n$:
14.5627674
79
$r_n$:
0.25348464
80
$\rho_n$:
14.4503043
80
$r_n$:
0.25153749
81
$\rho_n$:
14.3767803
81
$r_n$:
0.25026440
82
$\rho_n$:
14.2863118
82
$r_n$:
0.24869776
83
$\rho_n$:
14.2239571
83
$r_n$:
0.24761787
84
$\rho_n$:
14.1157902
84
$r_n$:
0.24574442
85
$\rho_n$:
14.0452618
85
$r_n$:
0.24452275
86
$\rho_n$:
13.9626271
86
$r_n$:
0.24309125
87
$\rho_n$:
13.8849703
87
$r_n$:
0.24174588
88
$\rho_n$:
13.7904978
88
$r_n$:
0.24010903
89
$\rho_n$:
13.7120949
89
$r_n$:
0.23875048
90
$\rho_n$:
13.6208737
90
$r_n$:
0.23716968
91
$\rho_n$:
13.5633748
91
$r_n$:
0.23617319
92
$\rho_n$:
13.4878634
92
$r_n$:
0.23486444
93
$\rho_n$:
13.4258226
93
$r_n$:
0.23378908
94
$\rho_n$:
13.3486302
94
$r_n$:
0.23245100
95
$\rho_n$:
13.2858097
95
$r_n$:
0.23136197
96
$\rho_n$:
13.2112887
96
$r_n$:
0.23007002
97
$\rho_n$:
13.1421391
97
$r_n$:
0.22887110
98
$\rho_n$:
13.0644481
98
$r_n$:
0.22752399
99
$\rho_n$:
12.9972792
99
$r_n$:
0.22635924
100
$\rho_n$:
12.9360973
100
$r_n$:
0.22529825
101
$\rho_n$:
12.8693268
101
$r_n$:
0.22414026
102
$\rho_n$:
12.8065481
102
$r_n$:
0.22305144
103
$\rho_n$:
12.7396985
103
$r_n$:
0.22189193
104
$\rho_n$:
12.6710008
104
$r_n$:
0.22070029
105
$\rho_n$:
12.6206479
105
$r_n$:
0.21982682
106
$\rho_n$:
12.5580705
106
$r_n$:
0.21874122
107
$\rho_n$:
12.4984677
107
$r_n$:
0.21770716
108
$\rho_n$:
12.4268412
108
$r_n$:
0.21646443
109
$\rho_n$:
12.3823879
109
$r_n$:
0.21569312
110
$\rho_n$:
12.3000527
110
$r_n$:
0.21426442
111
$\rho_n$:
12.2463575
111
$r_n$:
0.21333263
112
$\rho_n$:
12.1906904
112
$r_n$:
0.21236658
113
$\rho_n$:
12.1475148
113
$r_n$:
0.21161727
114
$\rho_n$:
12.0965651
114
$r_n$:
0.21073300
115
$\rho_n$:
12.0509312
115
$r_n$:
0.20994095
116
$\rho_n$:
11.9886435
116
$r_n$:
0.20885980
117
$\rho_n$:
11.9433898
117
$r_n$:
0.20807428
118
$\rho_n$:
11.8858176
118
$r_n$:
0.20707488
119
$\rho_n$:
11.8437245
119
$r_n$:
0.20634415
120
$\rho_n$:
11.7866487
120
$r_n$:
0.20535328
121
$\rho_n$:
11.7339187
121
$r_n$:
0.20443781
122
$\rho_n$:
11.6770715
122
$r_n$:
0.20345081
123
$\rho_n$:
11.6364348
123
$r_n$:
0.20274524
124
$\rho_n$:
11.5887834
124
$r_n$:
0.20191783
125
$\rho_n$:
11.5384928
125
$r_n$:
0.20104456
126
$\rho_n$:
11.4894028
126
$r_n$:
0.20019210
127
$\rho_n$:
11.4535231
127
$r_n$:
0.19956901
128
$\rho_n$:
11.4068507
128
$r_n$:
0.19875847
129
$\rho_n$:
11.3563380
129
$r_n$:
0.19788121
130
$\rho_n$:
11.3165625
130
$r_n$:
0.19719039
Definition
The covering radius is the greatest distance to the nearest of $n$ cap centres on the unit sphere $S^2$ [6]. Each selected configuration is listed in two invertible normalisations: angular radius $\rho_n$ in degrees and chord radius $r_n$.
Parameters
$n$
—   number of caps ($n\geq1$)
normalisation
—   normalisation of the covering radius (either angle or chord)
Formulas
(1)
$\displaystyle \rho_n=\frac{180}{\pi}\max_{y\in S^2}\min_{x\in X_n}\arccos\langle x,y\rangle$, where $X_n$ is the selected configuration of cap centres.
(2)
$r_n=2\sin(\pi\rho_n/360)$ and $\rho_n=(360/\pi)\arcsin(r_n/2)$, with $0<\rho_n\leq180$ and $0<r_n\leq2$.
(3)
If the origin lies inside $\operatorname{conv}(X_n)$ and $h_n$ is the smallest distance from the origin to a supporting facet plane, then $\rho_n=(180/\pi)\arccos(h_n)$ and $r_n=\sqrt{2-2h_n}$.
(4)
For $n=10,14$, put $m=(n-2)/2$ and $c=\cos(\pi/m)$. An optimal configuration consists of the north and south poles and two staggered regular $m$-gons at heights $\pm c/(1+c)$. Its facet distance satisfies $h_n^2=c^2(1+2c)/(1+c^2(1+2c))$ [4].
Comments
(5)
The numerical snapshot is the table of coverings by $n$ spherical caps and the files cov3-n.txt in the Hardin–Sloane–Smith collection [6], maintained by Henry Cohn and downloaded on 25-Oct-2026. The configurations were computed in 1994. The archive table rounds covering radii upward to seven decimal places; this differs from nearest rounding in some entries of the older page [7]. The attached source file preserves every listed radius, all coordinates, and hashes of the original files.
(6)
The angle $\rho_n$ is a cap radius, not its diameter or the minimum separation between cap centres. Its radian measure is $\pi\rho_n/180$. The chord radius $r_n$ is the straight-line distance between a cap centre and a boundary point on the unit sphere. These are two invertible normalisations of one covering radius.
(7)
The cases $n=1,2,3$ have elementary optimality proofs. Optimality is proven for $n=4,6,12$ [1], for $n=5,7$ [2], for $n=8$ [3], and for $n=10,14$ [4]. The cited proofs and their scope are also recorded in the survey [5], except for the later proof for $n=8$. For the other listed values of $n$, this table makes no claim of proven global optimality.
(8)
The source-derived entries describe the configurations in the cited snapshot. Their numerical uncertainty does not bound how much a better covering might improve the radius. Among the source configurations, exact geometric formulas are used for $n=4,5,6,7,10,12,14$; the proven case $n=8$ is retained at numerical source precision.
(9)
The source begins at $n=4$; the cases $n=1,2,3$ are supplied by elementary geometry. A single cap must reach the antipode of its centre. Any set of at most three centres lies in a closed hemisphere, so some point is at angular distance at least $90$ degrees from every centre. Two antipodal centres attain this bound, and adding any third distinct centre preserves the covering.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
R = RealBallField(numberdb.bits(100, losing=80))
# Covering by four caps centred at a regular tetrahedron.
h = R(1)/3
rho = 180*h.arccos()/R.pi()
r = (2-2*h).sqrt()
rho, r
References
[1]
L. Fejes Tóth, On covering a spherical surface with equal spherical caps (in Hungarian), Matematikai és Fizikai Lapok 50 (1943), 40–46.
[2]
K. Schütte, Überdeckungen der Kugel mit höchstens acht Kreisen, Mathematische Annalen 129 (1955), 181–186. (doi)
[3]
L. Wimmer, Covering the Sphere with Equal Circles, Discrete & Computational Geometry 57 (2017), 763–781. (doi)
[4]
G. Fejes Tóth, Kreisüberdeckungen der Sphäre, Studia Scientiarum Mathematicarum Hungarica 4 (1969), 225–247.
[5]
G. Fejes Tóth, Packing and covering, in J. E. Goodman, J. O’Rourke and C. D. Tóth (eds.), Handbook of Discrete and Computational Geometry, third edition, CRC Press (2018), 27–66, Table 2.4.1.
Links
Similar tables
Best known packings of equal circles in a circle —   maximises separation in a planar disk, another finite extremal configuration indexed by the number of centres
Best known packings of equal circles in an equilateral triangle —   maximises separation in a triangular planar container
Global minimum energies of Lennard-Jones clusters —   minimises pair-potential energy for a finite configuration of atoms
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds both normalisations for every $1\leq n\leq130$, combining the elementary cases $n=1,2,3$ with the complete contiguous range of unrestricted spherical coverings in the cited snapshot; coverings constrained to have icosahedral symmetry are a different optimisation problem and are not included)
Sources of data: [6], [7]
How they were obtained:

The seven source configurations $n=4,5,6,7,10,12,14$ are evaluated from exact geometric formulas to 100 significant digits in Sage real ball arithmetic with 80 guard bits, and repeated at 160 digits. Their minimum measured relative accuracy is 408 bits; the largest initial ball radius is less than $4.64\cdot10^{-122}$. The geometric derivations are given in the attached notes. All other source angles retain the source's seven decimal places, corresponding to nine significant digits in this range. Their chord radii are rounded to eight significant digits, with every written interval containing the full conversion of the source interval $[\rho_{\mathrm{printed}}-10^{-7},\rho_{\mathrm{printed}}]$ degrees.

more

Every one of the 127 published configurations was checked after radial normalisation. SciPy selects its convex-hull facets; explicit vector products recompute all 16,510 facet distances in real ball arithmetic at 192 and 320 bits. Both precisions agree, and every coordinate-derived angular radius lies within the source's upward-rounding interval. The largest discrepancy from the printed radius is less than $9.997\cdot10^{-8}$ degrees. The seven exact formulas also agree with the printed source values to this precision. The high precision of those rows comes from the exact formulas, not from the approximate coordinates.

Independent controls construct a tetrahedron, triangular and pentagonal bipyramids, an octahedron, an icosahedron, and the two polar configurations. Their facet radii agree with separately expressed closed forms; a deliberately perturbed tetrahedron is distinguished from the regular one. Both normalisation conversions, monotonicity across the range, the cap-area lower bound, hull incidence and Euler's identity are checked. The attached checks can also read all values from the database. These computations establish numerical agreement, not global optimality; hull candidate selection remains a numerical step.

For $n=1,2,3$, the angular radii and the chord radius for $n=1$ are exact integers. The chord radii for $n=2,3$ are computed as $\sqrt2$ in real ball arithmetic to 100 significant digits and repeated at 160 digits. Exact rational squaring verifies that each stored decimal interval contains $\sqrt2$; an independent interval trigonometric conversion agrees. The attached notes give the elementary optimality proofs.



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