Annulus Area: worked example
| Outer radius | 10 |
|---|---|
| Inner radius | 6 |
| Ring area | 201.1 |
| Ring width | 4 |
| Combined edge length | 100.5 |
Annulus Area: formula
Enter outer radius R and inner radius r in the same length unit, with 0 ≤ r ≤ R. For a washer measured by diameters, halve both first. The combined edge length includes both the inner and outer boundaries; the material face is the region between them.
Annulus Area: changing Outer radius
| Outer radius | Ring area | Ring width | Combined edge length |
|---|---|---|---|
| 8 | 88 | 2 | 88 |
| 9 | 141.4 | 3 | 94.2 |
| 10 (example) | 201.1 | 4 | 100.5 |
| 11 | 267 | 5 | 106.8 |
| 12 | 339.3 | 6 | 113.1 |
Annulus Area: compare input ranges
| Input | Tested from | To | Ring area: low input | Ring area: high input | Swing |
|---|---|---|---|---|---|
| Outer radius | 9 | 11 | 141.4 | 267 | 125.7 (63%) |
| Inner radius | 5.4 | 6.6 | 222.6 | 177.3 | 45.2 (23%) |
Annulus Area: individual input comparisons
| Input | Value | Ring area | Ring width | Combined edge length |
|---|---|---|---|---|
| Outer radius | 8 | 88 | 2 | 88 |
| 9 | 141.4 | 3 | 94.2 | |
| 10 | 201.1 | 4 | 100.5 | |
| 11 | 267 | 5 | 106.8 | |
| 12 | 339.3 | 6 | 113.1 | |
| Inner radius | 4.8 | 241.8 | 5.2 | 93 |
| 5.4 | 222.6 | 4.6 | 96.8 | |
| 6 | 201.1 | 4 | 100.5 | |
| 6.6 | 177.3 | 3.4 | 104.3 | |
| 7.2 | 151.3 | 2.8 | 108.1 |
Annulus Area: combined input changes
| Outer radius / Inner radius | 4.8 | 6 | 7.2 |
|---|---|---|---|
| 8 | 128.7 | 88 | 38.2 |
| 9 | 182.1 | 141.4 | 91.6 |
| 10 | 241.8 | 201.1 | 151.3 |
| 11 | 307.8 | 267 | 217.3 |
| 12 | 380 | 339.3 | 289.5 |
Annulus Area: which assumptions move the result?
Annulus Area: values and definitions
| Quantity and definition | Value used |
|---|---|
| Outer radius | 10 |
| Inner radius | 6 |
| Ring area | 201.1 |
| Ring width | 4 |
| Combined edge length | 100.5 |
Common error: Multiplying the ring width by the circumference of the outer circle. That ignores the curve of the ring and overstates the area, more so as the ring gets wider.
Concentric circles define an annulus. Subtracting inner-circle area from outer-circle area also gives the remaining area of an off-center circular hole if the hole stays entirely inside, but there is no single uniform ring width then. Overlapping boundaries need a circle-intersection calculation.
Geometry calculators: Regular Polygon Calculator, Rhombus Area Calculator, Hemisphere Volume Calculator, Cone Frustum Calculator, Capsule Volume Calculator, Triangular Prism Volume Calculator.
A ring's area is the big disc minus the hole, and the difference grows faster than the ring is wide. An outer radius of 10 with an inner radius of 6 leaves a ring 4 wide with an area of about 201 square units. A 4 wide ring near the center of a small disc would hold far less.
That is why a washer, a pipe wall and a running track all need the two radii, not just the width.
Measure both radii from the same center. If you have diameters, halve them first, and make sure the inner value is the smaller one.
See the circle area calculator for each circle, the cylinder page for tubes, and the arc length and sector calculator for parts of rings. The sphere, hemisphere and capsule pages cover round solids, and the unit conversion calculators change units.
Is width times average circumference approximate?
For concentric circles it is exact: (R − r) × π(R + r) = π(R² − r²).
How do I find the area of a ring?
Subtract the area of the inner circle from the area of the outer circle: pi times the outer radius squared minus the inner radius squared.
Where do annuli appear in practice?
Washers, pipes and tubes in cross-section, rings, tracks, gaskets and the shaded band around a circular object.
Is the ring's area equal to the width times its circumference?
Exactly, using the mean circumference: width times the mean circumference of the two circles gives the exact area of the ring.