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A crescent, or lune, is the moon-shaped region left over between two circular arcs that share the same chord and the same central angle but differ in radius — geometrically, it's what remains after subtracting a smaller circular segment from a larger one that spans the same opening. This particular model treats the crescent as the area difference between an outer arc's circular segment (radius R) and an inner arc's circular segment (radius r), both cut by the same central angle θ. Crescent-shaped area calculations show up in architectural and decorative design (window tracery, ornamental ironwork, and Islamic geometric patterns frequently use lune shapes), in the classical geometry problem known as the "lune of Hippocrates" (one of the earliest curved shapes ever proven to have an exactly computable area equal to a straight-edged triangle), and in general in any layout or manufacturing task that needs the area of a crescent cutout, such as a washer-like part with a curved rather than straight inner edge.
How it works
A circular segment is the region between a chord and the arc it cuts off, and its area is given by (θ − sinθ)/2 × R², where θ is the central angle in radians. Enter the outer radius R, the inner radius r, and the central angle θ they share; the calculator computes each circle's segment area using that formula and subtracts the smaller (inner) segment from the larger (outer) one, which simplifies algebraically to (θ − sinθ)/2 × (R² − r²) since both segments share the same angle term.
- Enter outer arc radius (R).
- Enter inner arc radius (r).
- Enter shared central angle (degrees).
- Click Calculate to see your results.
Examples
R=10, r=6, θ=90°
Converting 90° to radians gives π/2 ≈ 1.5708. (θ − sinθ)/2 = (1.5708 − 1)/2 ≈ 0.2854. Multiplying by (R² − r²) = (100 − 36) = 64 gives an area of about 0.2854 × 64 ≈ 18.27.
R=8, r=5, θ=60°
Converting 60° to radians gives π/3 ≈ 1.0472, and sin(60°) ≈ 0.866. (θ − sinθ)/2 = (1.0472 − 0.866)/2 ≈ 0.0906. Multiplying by (64 − 25) = 39 gives an area of about 0.0906 × 39 ≈ 3.53.
R=12, r=12, θ=45° (equal radii)
When R equals r, the term (R² − r²) becomes 0, so the crescent area is exactly 0 — geometrically, the "outer" and "inner" arcs coincide exactly, leaving no region between them.
Common mistakes to avoid
- Entering an inner radius larger than the outer radius, which produces a negative "crescent" area that has no physical meaning in this model.
- Forgetting to convert the angle to radians before applying the (θ − sinθ) formula, since sinθ itself expects θ in radians for the subtraction to be mathematically consistent.
- Confusing this simplified two-arc, shared-chord crescent with the more elaborate lune of Hippocrates construction, which involves a right triangle and different geometric relationships.
- Assuming the crescent area scales linearly with the radius difference (R − r) — it actually depends on the difference of squares (R² − r²), which grows faster for larger radii.