Given a unit disk, find the smallest radius required for equal disks to completely cover the unit
disk. The first few such values are

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

Here, values for ,
8, 9, 10 are approximate values obtained using computer experimentation by Zahn (1962).

For a symmetrical arrangement with (known as the five disks
problem), ,
where
is the golden ratio. However, rather surprisingly,
the radius can be slightly reduced in the general disk covering problem where symmetry
is not required; this configuration is illustrated above (Friedman). Neville (1915)
showed that the value is equal to , where and are solutions to

(11)

(12)

(13)

(14)

These solutions can be found exactly as

(15)

(16)

where

(17)

(18)

are the smallest positive roots of the given polynomials, with denoting the th root of the polynomial in the ordering of the Wolfram
Language. This gives (OEIS A133077)
exactly as

(19)

where the root is the smallest positive one of the above polynomial.

is also given by ,
where
is the largest real root of

(20)

maximized over all , subject to the constraints

(21)

(22)

and with

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(Bezdek 1983, 1984).

Letting
be the smallest number of disks of radius needed to cover a disk , the limit of the ratio of the area
of
to the area of the disks is given by

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