How this instrument works
When a smaller circle of radius r sits entirely within a larger circle of radius R, the crescent-shaped area left behind — the larger circle's area minus the smaller one's — is simply A = π(R² − r²), the straightforward difference between the two circles' own areas. No special crescent-specific formula is actually needed; the shape's area is just what's left once the smaller circle's contribution is subtracted from the larger.
This is the same 'bite taken out of a circle' geometry behind a washer's face (a flat ring with a hole, viewed from one perspective) or a crescent moon's simplified geometric model — a shape wider than the removed piece, with the two boundary curves both circular but at different radii.
This formula assumes the smaller circle sits fully inside the larger one, sharing no particular alignment requirement (it doesn't need to be centered) as long as it doesn't cross the larger circle's own boundary — if the smaller circle instead only partially overlaps the larger one, or pokes outside it, the leftover area takes a different, more involved shape entirely, needing circular-segment geometry rather than this simple subtraction.
- Enter the larger circle's radius into the Larger circle radius field.
- Enter the smaller circle's radius (assumed fully inside the larger one) into the Smaller circle radius field.
- Read Crescent area: the sheet subtracts the smaller circle's area from the larger one's directly.
Worked example — radii 5 and 3
A larger circle of radius 5 has a smaller circle of radius 3 removed from inside it. The crescent area left over is π(25−9) = 16π ≈ 50.27 — simply the difference between the two circles' own areas, with no additional geometry needed.
A smaller circle of radius 0 removes nothing at all, leaving the full larger circle's own area, 25π. Equal radii (5 and 5), by contrast, leave no crescent whatsoever — the smaller circle exactly fills the larger one, and the remaining area is exactly zero.
Questions
What is the formula for a crescent's area?
A = π(R² − r²), where R is the larger circle's radius and r is the smaller circle's radius, assuming the smaller circle sits entirely inside the larger one. It's simply the difference between the two circles' own areas.
Does the smaller circle need to be centered inside the larger one?
No — as long as the smaller circle sits entirely within the larger circle's boundary without crossing it, this formula holds regardless of exactly where inside the larger circle it sits.
What if the smaller circle only partially overlaps the larger one?
This formula no longer applies — a partially overlapping pair of circles produces a lens-shaped overlap region requiring circular-segment geometry, a different and more involved calculation than the simple area subtraction used here.
What if the two radii are equal?
The crescent area is exactly zero — the smaller circle exactly fills the larger one, leaving nothing behind.
Is this the same shape as a crescent moon?
It's the simplified geometric model of one — a genuine crescent moon's shape comes from sunlight illuminating part of a sphere as seen from a particular angle, a more complex 3D phenomenon, but the flat, two-circles version captures the same basic 'wider curve minus narrower curve' idea.