An annulus is the ring-shaped region between two concentric circles — the same centre, two different radii. Real-world annuli include washers, the cross-section of a pipe, a circular path or track, and a CD. This calculator takes the outer and inner radius and returns the ring’s area, its width, and the circumference of both the outer and inner edges.
How it works
Let R be the outer radius and r the inner radius (the radius of the hole). The tool uses these standard formulas:
- Ring area = π(R² − r²) — the large circle’s area minus the small circle’s area.
- Ring width = R − r — the radial thickness of the band.
- Outer circumference = 2πR.
- Inner circumference = 2πr.
Enter both radii in the same unit and every result comes back in matching units (area in square units).
Worked example
A washer with an outer radius R = 10 mm and inner radius r = 6 mm:
- Area = π(10² − 6²) = π(100 − 36) = 64π ≈ 201.06 mm²
- Width = 10 − 6 = 4 mm
- Outer circumference = 2π × 10 ≈ 62.83 mm
- Inner circumference = 2π × 6 ≈ 37.70 mm
| Quantity | Formula | Result |
|---|---|---|
| Ring area | π(R² − r²) | 201.06 mm² |
| Ring width | R − r | 4 mm |
| Outer circumference | 2πR | 62.83 mm |
| Inner circumference | 2πr | 37.70 mm |
Where annulus calculations come up
Engineering and manufacturing. Pipe cross-sections, bearing races, and O-ring grooves all have annular geometry. The ring area directly tells you material usage or flow area. For hollow structural sections, the area is also the basis of the section’s load-bearing capacity.
Architecture and landscaping. A circular fountain with a central island, a running track around an infield, or a donut-shaped planter all have annular areas. Multiply the ring area by a depth to get excavation or fill volume.
Optics. An annular aperture — for example the beam path around a central obstruction in a Cassegrain telescope — is an annulus. The ratio of obstructed area to total aperture area describes the fraction of light lost.
Common mistakes and edge cases
- Inner radius must be smaller than outer radius. If r ≥ R the ring makes no sense; most tools flag this as an error.
- Units must match. If R is in centimetres and r is in millimetres, convert first. The area unit is always the square of whichever linear unit you used.
- Working from diameters? Halve each diameter to get the radius (R = D/2, r = d/2) before entering. The tool accepts radii, not diameters.
- Near-equal radii produce thin rings. As r approaches R the area goes toward zero, which is correct — a ring of negligible width has negligible area.