An annulus (plural: annuli) is the ring-shaped region between two concentric circles — two circles that share the same centre but have different radii. The area of that ring is one of the most frequently needed measurements in engineering, manufacturing, architecture and everyday maths. This calculator handles three common situations: find the ring area from both radii, find the outer radius from the area and the inner radius, or find the inner radius from the area and the outer radius.
The core formula
Let R be the outer radius and r be the inner radius. The ring area is simply the area of the large circle minus the area of the small circle:
A = π(R² − r²)
Because both circles are concentric the “missing middle” subtracts cleanly, leaving the shaded band. The same result can be written as A = π(R + r)(R − r), which is sometimes useful for mental arithmetic: multiply the sum of the radii by their difference, then multiply by π.
Two bonus measurements follow for free once you have R and r:
- Ring width = R − r
- Outer circumference = 2πR
- Inner circumference = 2πr
Solving for a missing radius
Real problems do not always hand you both radii. If a designer specifies a required ring area and one of the radii, you need to back-calculate the other:
- Outer radius from area and inner radius: R = sqrt(A / π + r²)
- Inner radius from area and outer radius: r = sqrt(R² − A / π)
Both are exact algebraic rearrangements of the same formula. This calculator’s “Solve for…” menu switches between all three modes automatically, relabelling the input fields so you always know which two values to provide.
Worked example
A stainless-steel washer has an outer radius of 18 mm and an inner radius of 7 mm.
- R² = 18² = 324
- r² = 7² = 49
- R² − r² = 324 − 49 = 275
- A = π × 275 ≈ 863.94 mm²
- Ring width = 18 − 7 = 11 mm
- Outer circumference = 2π × 18 ≈ 113.10 mm
- Inner circumference = 2π × 7 ≈ 43.98 mm
Now suppose a different washer must have a ring area of 500 mm² with an inner radius of 8 mm. What outer radius is needed?
- A / π = 500 / π ≈ 159.155
- A / π + r² = 159.155 + 64 = 223.155
- R = sqrt(223.155) ≈ 14.94 mm
Both calculations are available in the tool above by switching the “Solve for…” dropdown.
| Outer R | Inner r | Ring area | Width |
|---|---|---|---|
| 10 mm | 6 mm | 201.06 mm² | 4 mm |
| 18 mm | 7 mm | 863.94 mm² | 11 mm |
| 50 cm | 30 cm | 5026.55 cm² | 20 cm |
| 1 m | 0.75 m | 1.3744 m² | 0.25 m |
Formula note
The identity A = π(R² − r²) = π(R + r)(R − r) is an exact result from Euclidean geometry requiring only the two radii. No approximations are made — the only non-integer constant is π, which JavaScript represents internally as the IEEE 754 double-precision value 3.141592653589793. The decimal-places selector controls display rounding only; the arithmetic uses the full floating-point precision throughout.
Frequently asked questions
Why is an annulus also called a ring or circular ring? The word “annulus” comes from the Latin for “little ring.” In everyday language the shape is often described as a ring, washer shape, or hollow circle — all referring to the same concentric-circle region.
What happens if inner radius equals outer radius? If r = R the ring has zero width and zero area. The calculator will flag this as an error because R must be strictly greater than r for a valid annulus.
Can the inner radius be zero? Yes — when r = 0 the annulus degenerates into a full disc and the formula reduces to A = πR², which is the standard circle area formula. The tool accepts r = 0 and computes the correct disc area.
What are typical annulus area applications? Engineers use it to find the cross-sectional area of hollow cylinders and pipes (which directly determines flow rate or stress). Architects use it for circular running tracks, fountain basins and curved pavements. Manufacturers use it for washer and gasket blanking, and laser-cutting path planning.