Annulus Calculator
Calculate area and perimeter of an annulus — the ring between two concentric circles.
Enter outer and inner radius with unit select for instant results.
Annulus (Ring Shape)
An annulus is the flat region between two concentric circles — a larger outer circle and a smaller inner circle sharing the same center. Think of a washer, a ring donut cross-section, a CD/DVD, or a circular frame.
Formulas:
| Property | Formula |
|---|---|
| Area | A = pi * (R^2 - r^2) |
| Outer Circumference | C_out = 2 * pi * R |
| Inner Circumference | C_in = 2 * pi * r |
| Total Perimeter | P = 2 * pi * (R + r) |
| Width (ring thickness) | w = R - r |
Variables:
- R = outer radius (larger circle)
- r = inner radius (smaller circle, must be less than R)
Useful alternative form: A = pi * (R + r) * (R - r) = pi * (R + r) * w
This shows that annulus area depends on both the mean radius and the ring width.
Worked example — washer (R = 5 cm, r = 3 cm):
- Area: pi * (25 - 9) = 16pi ≈ 50.27 cm^2
- Outer perimeter: 2pi × 5 ≈ 31.42 cm
- Inner perimeter: 2pi × 3 ≈ 18.85 cm
- Total perimeter: 2pi × 8 ≈ 50.27 cm
- Ring width: 5 - 3 = 2 cm
Interesting note: An annulus with R = 1 m has the same area (pi m^2) as a circle with radius 1 m, but only when r = 0 (degenerate case). As r approaches R, the area approaches zero and the shape becomes an infinitely thin ring.
Real-world annuli: Pipe cross-sections, washers, rings, circular frames, road roundabouts, and the rings of Saturn all form annular shapes. The annulus also appears in probability: the area of a circular ring represents the probability of a random point landing at a given radial distance in a circle.