How Many Small Circles Fit in a Big One
Dividing areas straight across overcounts. However neatly you arrange circles, gaps remain: even on an infinite plane the best hexagonal packing reaches only 90.69%. This multiplies by that limit.
Your numbers
How many fit
22
Area ratio
25×
Area
706.9cm²
What it means
About 22 fit. The area ratio suggests 25, but gaps between circles leave only 90.7% usable.
Formula
How many fit = 0.9069 × (Diameter ÷ Small circle diameter) ²
For small counts — under ten or so — the error is large. Edge effects usually mean fewer fit than this suggests.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Diameter (cm)The longest line across a circle — twice the radius. | 30 | 0 and up |
| Small circle diameter (cm)The diameter of the small circles being packed in. | 6 | 0.01 and up |
Step by step
Quick reference table
Results when only Diameter (cm) changes and everything else stays put.
| Diameter (cm) | How many fit | Area ratio (×) | Area (cm²) |
|---|---|---|---|
| 15 | 5 | 6.25 | 176.7 |
| 22.5 | 12 | 14.06 | 397.6 |
| 30 | 22 | 25 | 706.9 |
| 45 | 51 | 56.25 | 1,590.4 |
| 60 | 90 | 100 | 2,827.4 |
What each result means
| Result | At default values |
|---|---|
| How many fitHow many actually fit once the gaps are counted. | 22 |
| Area ratio (×)How much area grows when length scales by k — k squared. | 25 |
| Area (cm²)The space a shape covers — length squared. | 706.9 |
Common mistakes
For small counts — under ten or so — the error is large. Edge effects usually mean fewer fit than this suggests.
Glossary
- Diameter
- The longest line across a circle — twice the radius.
- Small circle diameter
- The diameter of the small circles being packed in.
- How many fit
- How many actually fit once the gaps are counted.
- Area ratio
- How much area grows when length scales by k — k squared.
- Area
- The space a shape covers — length squared.
Frequently asked questions
Q. How is How Many Small Circles Fit in a Big One calculated?
How many fit = 0.9069 × (Diameter ÷ Small circle diameter) ² — Dividing areas straight across overcounts. However neatly you arrange circles, gaps remain: even on an infinite plane the best hexagonal packing reaches only 90.69%. This multiplies by that limit.
Q. Can you walk through an example?
With Diameter 30cm, Small circle diameter 6cm, the answer is How many fit 22.
Q. What do I need to enter?
Enter Diameter, Small circle diameter. The result recalculates as you type, and an empty box counts as zero.
Q. How much does the answer move if I change a number?
Changing only Diameter (cm) moves the answer to Diameter (cm) 15 → How many fit 5 and Diameter (cm) 60 → How many fit 90. The table below lays out five steps.
Q. How are the numbers rounded?
Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.
Q. Anything to watch out for?
For small counts — under ten or so — the error is large. Edge effects usually mean fewer fit than this suggests.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.