Ring area
A = π(R² − r²)
The equivalent factored form A = π(R − r)(R + r) = πw(R + r) is especially useful for thin rings.
Annulus Calculator — Ring Area, Circumference and Radii
Enter Two Known Values
Annulus Results
Outer radius (R)—
Inner radius (r)—
Ring width (w)—
Ring area (A)—
Outer diameter—
Inner diameter—
Outer circumference—
Inner circumference—
Total boundary length—
Enter two independent measurements, then calculate to see the formula path.
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Annulus Formulas
An annulus is the region between two concentric circles. Let R be the outer radius, r the inner radius, and w = R − r the radial width.
Outer and inner circumference
Cₒ = 2πR
Cᵢ = 2πr
Total boundary length
B = Cₒ + Cᵢ = 2π(R + r)
This counts both circular edges of the ring.
Radii from width and area
R + r = A/(πw)
Then R = ½(A/(πw) + w) and r = ½(A/(πw) − w).
Assumption: The two circles share the same centre and use one common length unit. The calculator requires
R > r ≥ 0. An inner radius of zero is shown as the limiting solid-disk case.
How to Use the Annulus Calculator
- Select two independent measurements. Radius and its matching circumference are equivalent, so they cannot solve a ring by themselves.
- Enter the values and choose their shared length unit. Ring area uses the corresponding square unit automatically.
- Select Calculate annulus to obtain radii, width, area, diameters, and all boundary lengths.
- Copy the formatted summary or download unrounded values as a CSV file if needed.
Changing the unit selector changes the unit label; it does not convert values already entered. Convert both known measurements first if they originally use different units.
Worked Annulus Examples
Outer radius 10 cm, inner radius 5 cm
A = π(10² − 5²) = 75π ≈ 235.619 cm².
Cₒ = 20π ≈ 62.832 cm and Cᵢ = 10π ≈ 31.416 cm.
Width 3 m, area 51π m²
R + r = 51π/(3π) = 17 m.
With R − r = 3, the radii are R = 10 m and r = 7 m.
Outer radius 8 ft, area 39π ft²
r = √(R² − A/π) = √(64 − 39) = 5 ft.
The ring width is 8 − 5 = 3 ft.
Common Annulus Calculation Mistakes
Subtracting radii instead of squared radii
Area is π(R² − r²), not π(R − r). The difference R − r is only the ring width.
Using only one circumference
The outer and inner edges have different circumferences. Use their sum when a problem asks for the entire boundary length.
Mixing units
Convert every length to one unit before calculating. Area conversion factors are squared: 1 m² = 10,000 cm².
Annulus Calculator FAQs
What is an annulus?
An annulus is the ring-shaped region between two concentric circles. Its outer radius must be greater than its inner radius.
How do you calculate the area of an annulus?
Subtract the inner circle area from the outer circle area: A = π(R² − r²).
What circumference does an annulus have?
An annulus has an outer circumference Cₒ = 2πR and an inner circumference Cᵢ = 2πr. Their sum is the total boundary length.
How do you find annulus area from width and radii?
Because R² − r² = (R − r)(R + r), use A = πw(R + r), where w = R − r.
Can ring width and area determine both radii?
Yes. First find R + r = A/(πw). Combine that with R − r = w to find each radius.
Can the inner radius be zero?
Yes. The calculator allows an inner radius of zero as the limiting case, which produces a solid disk rather than a ring with a hole.
Are my annulus measurements uploaded?
No. The calculation runs locally in your browser and does not send your measurements to a server.
Calculation Notes
Last reviewed: July 30, 2026 by the Starlight Tools editorial team.
The calculator uses standard Euclidean circle relationships and JavaScript's full floating-point value of π. It evaluates area as πw(R + r) after solving, which avoids subtracting two nearly equal squared radii. Values outside reliable browser-number ranges or rings too thin to distinguish at the supplied magnitude are rejected with an explanation.