Circle Overlap Area
The overlap is a lens made of two circular segments. The law of cosines gives a half-angle in each circle, and the two segment areas add up. Beyond the sum of the radii the circles miss each other; below the difference the smaller one sits entirely inside.
Your numbers
Area
55.2cm²
Overlap share
27.5%
Formula
Area = Radius² × (α − sinα cosα) + Inner radius² × (β − sinβ cosβ)
The share is measured against the smaller circle — 100% means it is completely covered.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Radius (cm)Centre to edge — half the diameter. | 10 | 0 and up |
| Inner radius (cm)The smaller of two radii when the ends differ. | 8 | 0 and up |
| Centre distance (cm)The distance between the two shaft centres. | 12 | 0 and up |
Step by step
Quick reference table
Results when only Radius (cm) changes and everything else stays put.
| Radius (cm) | Area (cm²) | Overlap share (%) |
|---|---|---|
| 5 | 3.26 | 4.2 |
| 7.5 | 23.45 | 13.3 |
| 10 | 55.2 | 27.5 |
| 15 | 136.6 | 67.9 |
| 20 | 201.06 | 100 |
What each result means
| Result | At default values |
|---|---|
| Area (cm²)The space a shape covers — length squared. | 55.2 |
| Overlap share (%)The overlap as a share of the smaller circle’s area. | 27.5 |
Common mistakes
The share is measured against the smaller circle — 100% means it is completely covered.
Glossary
- Radius
- Centre to edge — half the diameter.
- Inner radius
- The smaller of two radii when the ends differ.
- Centre distance
- The distance between the two shaft centres.
- Area
- The space a shape covers — length squared.
- Overlap share
- The overlap as a share of the smaller circle’s area.
Frequently asked questions
Q. How is Circle Overlap Area calculated?
Area = Radius² × (α − sinα cosα) + Inner radius² × (β − sinβ cosβ) — The overlap is a lens made of two circular segments. The law of cosines gives a half-angle in each circle, and the two segment areas add up. Beyond the sum of the radii the circles miss each other; below the difference the smaller one sits entirely inside.
Q. Can you walk through an example?
With Radius 10cm, Inner radius 8cm, Centre distance 12cm, the answer is Area 55.2cm².
Q. What do I need to enter?
Enter Radius, Inner radius, Centre distance. 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 Radius (cm) moves the answer to Radius (cm) 5 → Area (cm²) 3.26 and Radius (cm) 20 → Area (cm²) 201.06. 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?
The share is measured against the smaller circle — 100% means it is completely covered.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.