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Crescent and Circle-Overlap Area Calculator — result sheet
Calculate the overlap and the two lune-shaped remainder areas formed by two intersecting circles, and identify which lune is a crescent.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For intersecting circles, overlap = r₁² acos((d²+r₁²−r₂²)/(2dr₁)) + r₂² acos((d²+r₂²−r₁²)/(2dr₂)) − ½√((−d+r₁+r₂)(d+r₁−r₂)(d−r₁+r₂)(d+r₁+r₂)); lune₁ = πr₁² − overlap and lune₂ = πr₂² − overlap.
Two overlapping disks create a central lens and two complementary lune regions. This calculator validates the intersecting-circle geometry, evaluates the circular-segment expression with bounded inverse-cosine inputs, and explains whether each lune excludes its own circle center—the geometric condition used for a crescent classification.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Radius r₁ — units; minimum 1.0E-6; maximum 100000
- Radius r₂ — units; minimum 1.0E-6; maximum 100000
- Center distance d — units; minimum 1.0E-6; maximum 200000
Worked example
| Input | Value |
|---|---|
| Radius r₁ | 4 |
| Radius r₂ | 6 |
| Center distance d | 5 |
Overlap ≈ 29.30 square units; lune 1 ≈ 20.96 and is a crescent, while lune 2 ≈ 83.70 and is not.
Assumptions and limits
- The circles have positive radii and two proper intersection points: |r₁−r₂| < d < r₁+r₂.
- Areas are planar Euclidean areas and use the same length unit for both radii and the center distance.
- A crescent label means the corresponding lune does not contain that circle’s center; it is not a claim about a lunar phase or physical object.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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