Circles around a circle calculator

What this calculator does

Put a circle down and surround it with circles of the same size, each touching the middle one. Exactly six fit, with no room to spare. That is not a coincidence or a good approximation, it is exact: six equilateral triangles meet at a point, and the surrounding centres sit on the vertices of a regular hexagon.

Change the sizes and the answer stops being tidy. Small circles around a big one leave slack that is not enough for one more, and the leftover angle is the useful output, because it tells you whether a fractionally smaller surrounding circle would let another in.

The formula

Formulaeach surrounding circle subtends 2 × arcsin(r ÷ (R + r)) at the centre, so the count is floor(360° ÷ that angle)

Join the centre of the middle circle to the centre of a surrounding one and the distance is R + r, since they touch. The surrounding circle subtends a half-angle of arcsin(r ÷ (R + r)) at the centre, so it occupies twice that. Dividing 360° by that angle and rounding down gives how many fit. The count depends only on the ratio of the radii, so doubling both changes nothing.

Term	Meaning
R	Radius of the central circle.
r	Radius of each surrounding circle.
Subtended angle	The angle one surrounding circle blocks off at the centre, 2 × arcsin(r ÷ (R + r)).
Slack	The leftover angle after the whole number of circles has been placed. Less than one circle wide by definition.
Kissing number	How many non-overlapping unit spheres can touch one more of the same size. The answer when R = r.

The inputs explained

Field	What to enter
Central circle radius	The central circle. Set it to zero to ask how many circles fit around a point, which is two.
Surrounding circle radius	The surrounding circles, all the same size. Equal radii gives the kissing number case.

When to use it

Confirming the hexagon

Set both radii equal. The answer is six, the angle each takes is exactly 60° and the slack is zero. This is the two-dimensional kissing number, and the only case where the packing is tight with nothing to spare.

Laying out pipes, cables or planting

A bundle of small pipes around a central one, or a ring of plants around a tree, is the same question. The gap figure tells you the spacing between neighbours once they are spread evenly, which is usually what you actually need to mark out.

Checking whether one more will fit

If the leftover angle is close to the angle one circle takes, a small reduction in the surrounding radius lets another in. If it is close to zero, you are already tight.

Worked examples

Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.

How many circles fit around a circle?

The surrounding circles shrink down the rows, so more of them fit.

Central radius fixed at 10

Surrounding radius	Circles that fit	Angle each one takes	Angle left over
10	6	60.00°	0.00°
5	9	38.94°	9.52°
3	13	26.68°	13.10°
2	18	19.19°	14.61°
1	34	10.43°	5.32°
0.5	65	5.46°	5.18°

Equal radii is the exact case: six circles at 60° each, with nothing left over. Everything below it wastes some angle. At a radius of 5 nine circles fit with 9.52° spare, which is well short of the 38.94° another would need. Halving the surrounding radius roughly doubles the count once the circles are small, because the subtended angle is then close to proportional to r.

Does the size of the central circle matter?

Only the ratio of the two radii matters, so this is the same question asked from the other side.

Surrounding radius fixed at 1

Central radius	Circles that fit	Angle each one takes	Gap between neighbours when evenly spread
1	6	60.00°	0.000
2	9	38.94°	0.052
5	18	19.19°	0.084
10	34	10.43°	0.030
50	160	2.25°	0.003
100	317	1.13°	0.002

A ratio of 10 to 1 fits 34 circles, whether that is radius 10 around radius 1 as in this table or radius 100 around radius 10, because only the ratio matters. As the central circle grows the surrounding circles effectively pack along a straight line and the count approaches π × (R ÷ r): at a central radius of 100 that estimate gives 314 against the true 317.

Questions

Why exactly six when the circles are the same size?

Because the centres form equilateral triangles. Each surrounding circle subtends arcsin(1/2) = 30° either side of its centre line, so 60° in total, and six 60° sectors is exactly 360°. It is the one case that comes out whole.

What is the kissing number?

The same question for spheres: how many unit spheres can touch a central one without overlapping. In two dimensions it is 6, in three it is 12, in four 24, in eight 240 and in twenty-four 196,560. Most other dimensions are still only known between bounds.

Does this allow the surrounding circles to overlap each other?

No. Each one must touch the centre circle and not overlap its neighbours, which is what the angle division enforces.

Can I fit more by using different sizes?

Yes, and that is a different and much harder problem. This calculator assumes every surrounding circle is the same size.

Why is the answer not a whole number of circles exactly?

Because the subtended angle rarely divides 360° evenly. The leftover is reported as slack, and it is always less than the angle one more circle would need.

For the circle geometry itself, see circle and circle sector and arc. The higher-dimensional version of this question, and the machine-found arrangement in eleven dimensions, is in the piece on the kissing number.