Area of Crescent Calculator

Find the area, circumference, and maximum width of a crescent (lune) — the sliver shape left when a smaller circle is removed from a larger one, touching at a single point.

Quick Facts

Formula
A = π(R² − r²)
The crescent's area equals the outer circle's area minus the inner circle's area.
Shape
Two circles, internally tangent
The inner circle sits inside the outer one and touches it at exactly one point.

Your Results

Calculated
Crescent area (A)
-
π(R² − r²)
Outer circumference
-
2πR — perimeter of the outer circle
Inner circumference
-
2πr — perimeter of the removed circle
Maximum width
-
2(R − r) — thickest point of the crescent

Ready

Enter the outer radius R and inner radius r, then press Calculate.

Formula and method for Area of Crescent

A crescent (also called a lune) is the sliver shape left over when a smaller circle is removed from inside a larger circle so the two circles touch at exactly one point. This calculator applies the standard formula for that shape: subtract the area of the inner circle from the area of the outer circle.

How the calculation works

Given the outer radius R and the inner radius r (with r < R), the area of the crescent is A = π(R² − r²). This works because the outer circle has area πR², and removing a circle of radius r from inside it takes away exactly πr² of area — the position of the smaller circle inside the larger one does not change how much area is removed, only the two radii matter. The calculator also reports the circumference of each circle (2πR and 2πr) and the crescent's maximum width, 2(R − r), which is the thickest point of the sliver, measured on the side directly opposite where the two circles touch.

Common mistakes

  • Radius vs. diameter: both R and r must be radii (half the diameter). Entering a diameter where a radius is expected roughly doubles the result.
  • r must be smaller than R: if the inner radius equals or exceeds the outer radius, there is no crescent left to measure — check that you haven't swapped the two values.
  • Mixed units: area comes out in square units (cm², in²) while circumference and width stay in linear units (cm, in) — keep the unit selector matched to how you measured R and r.

Real-world applications

  • Astronomy and illustration use crescent geometry to describe the sunlit sliver of a moon or planet
  • Machine design and gasket cutting use crescent shapes for washers, seals, and offset cutouts
  • Architecture and landscaping use crescent beds, windows, and moldings where two arcs of different radii meet
  • Logo and icon design frequently use the crescent/lune shape, where the exact area matters for material or ink coverage

Frequently Asked Questions

What is the formula for the area of a crescent?
The area of a crescent (lune) formed by an outer circle of radius R and an inner circle of radius r, tangent to each other at one point, is A = π(R² − r²) — the outer circle's area minus the inner circle's area.
What's the difference between a crescent and a lune?
In everyday geometry the two terms are used interchangeably for this two-circle shape. In advanced geometry, "lune" can also refer to the classical Lune of Hippocrates, which is bounded by two arcs sharing a common chord rather than two tangent circles. This calculator uses the two-tangent-circle definition, which is the common textbook form of a crescent.
Why must the inner radius be smaller than the outer radius?
If r were equal to or greater than R, the inner circle would not fit entirely inside the outer circle, so there would be no crescent-shaped remainder to compute. The calculator requires r to be strictly less than R.
What is the maximum width of a crescent?
Measured along the line through both circle centers, on the side opposite where the two circles touch, the widest point of the crescent equals 2(R − r) — twice the difference between the outer and inner radii.

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