Circle Overlap Area Calculator (Area of Intersection of Two Circles)
Area of the lens where two circles overlap, from their radii and the distance between centres, with the share of each circle covered.
Results
The dashed line marks the value you entered.
What this tool does
Two overlapping circles share a lens-shaped region, and its area is the sum of two circular segments, one from each circle. The formula looks long, but it only needs the three lengths. There are two cases that need care: if the centres are farther apart than the two radii added up the area is zero, and if one circle sits wholly inside the other the overlap is the whole smaller circle. This tool handles both, and the chart shows how the overlap shrinks as the circles move apart. Two unit circles whose centres are 1 apart overlap by about 1.228.
Formula
A = r₁² cos⁻¹((d² + r₁² − r₂²) ÷ 2dr₁) + r₂² cos⁻¹((d² + r₂² − r₁²) ÷ 2dr₂) − ½ √((−d+r₁+r₂)(d+r₁−r₂)(d−r₁+r₂)(d+r₁+r₂))
Variables
| Symbol | Meaning | Unit |
|---|---|---|
r1 | Radius of circle 1 | — |
r2 | Radius of circle 2 | — |
d | Distance between the centres | — |
A | Overlapping area (the lens) | — |
S1 | Share of circle 1 that is covered | % |
S2 | Share of circle 2 that is covered | % |
U | Area covered by at least one circle | — |
CS | How the circles lie | — |
Worked example
- Radius of circle 15
- Radius of circle 24
- Distance between the centres6
- Overlapping area (the lens)13.7995
- Share of circle 1 that is covered17.57 %
- Share of circle 2 that is covered27.45 %
- Area covered by at least one circle115.0058
- How the circles lieThey partly overlap
Limitations
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.