The area of an annulus is π(R² − r²), where R is the outer radius and r is the inner radius of the ring. This single subtraction captures the whole idea behind a ring: take the area of the large outer disk (πR²) and cut out the small inner disk (πr²), leaving a flat band — the annulus itself. Because both circles share the same center, the geometry stays clean and the formula factors neatly into π(R² − r²). Engineers, machinists, landscapers, and geometry students meet this shape everywhere: the cross-section of a pipe wall, the load-bearing surface of a hollow shaft, the metal area of a washer, a circular running-track lane, or simply the ring around a CD or flat doughnut. The fastest way to compute it is with an Annulus Area Calculator: type the outer radius, type the inner radius, and the tool prints π(R² − r²) along with the outer area, inner area, and ring width so you can double-check every part of the result. The math itself is short, but the small details — units, the order of R and r, and what the answer actually represents — decide whether the number you get is correct.
What the Annulus Area Formula Actually Means
An annulus is the flat region between two concentric circles — two circles that share the same center but have different radii. The "concentric" part matters: it guarantees the inner circle sits neatly inside the outer one, with no overlap and no offset, so the difference of the two disks is itself a clean geometric shape. If the centers were not the same, you would not get a ring — you would get a crescent or a lens, and a different formula altogether.
Start with the area of a full circle, πr². If you draw one circle of radius R, its area is πR². Inside that circle, draw a second, smaller circle of radius r that shares the same center. The smaller circle has area πr². The annulus is the region left behind when you remove the smaller disk from the larger one. That gives the area as a subtraction:
Annulus area = πR² − πr² = π(R² − r²)
The factoring step is what makes the formula memorable. Instead of calculating two areas and subtracting them, you can square each radius once, subtract the smaller from the larger, and multiply by π. The shape itself is sometimes just called a ring, and in everyday language it shows up as a washer, a flat doughnut, the rim around a pond, or the wall of a pipe cut across its length. Anywhere you see a circular band with a hole in the middle, you are looking at an annulus and the same π(R² − r²) rule applies.
How to Get the Answer with the Annulus Area Calculator
The calculator removes the only fiddly part of the formula — remembering to keep the units consistent and the radii in the right order — so all you do is type two numbers and read the result.
- Open the Annulus Area Calculator in your browser. The page shows two input boxes — one for R and one for r.
- Type the outer radius (R) into the first box. Pick any unit you like: centimeters, meters, inches, feet, or millimeters — the calculator stays unit-agnostic and accepts the raw number.
- Type the inner radius (r) into the second box. Make sure r is smaller than R; if r equals R the ring collapses to a line and the area is exactly zero, while r larger than R is flagged as invalid rather than producing a meaningless negative area.
- Read the result panel below the boxes. It shows the annulus area as π(R² − r²), along with the outer circle's area (πR²), the inner circle's area (πr²), and the ring width (R − r).
- If you need to compare scenarios, adjust either radius and re-read the updated numbers. The calculation is instant because the math runs locally in your browser, with nothing uploaded to a server.
A Worked Example You Can Verify by Hand
Suppose R = 10 and r = 6, both in centimeters. Plug the numbers into π(R² − r²):
R² = 10² = 100 r² = 6² = 36 R² − r² = 100 − 36 = 64 Annulus area = π × 64 = 64π ≈ 201.06 square centimeters.
As a sanity check, you can also compute it as outer minus inner: the outer circle's area is πR² = 100π ≈ 314.16 cm², the inner circle's area is πr² = 36π ≈ 113.10 cm², and the difference is 100π − 36π = 64π ≈ 201.06 cm². The ring width is R − r = 10 − 6 = 4 cm. These three intermediate values — outer area, inner area, and ring width — are exactly what the calculator displays alongside the final annulus area, so you can confirm each step of the math without redoing the arithmetic by hand.
Where This Formula Shows Up in Real Work
An annulus is one of those shapes that quietly appears in many trades. The table below lists common situations where this calculation matters, the two radii you typically have on hand, and what the resulting area represents in practice.
| Field | Radii you know | What the area represents |
|---|---|---|
| Mechanical engineering | Outer pipe diameter and inner pipe diameter | Cross-sectional metal area of a hollow shaft, tube, or pipe wall |
| Fastener design | Outer washer diameter and inner hole diameter | Metal surface area of a washer or flange |
| Sealing and gaskets | Outer gasket diameter and inner gasket diameter | Material area of a ring-shaped gasket |
| Architecture and landscape | Outer ring radius and inner ring radius | Paving area of a circular path, pond rim, or running-track lane |
| Manufacturing | Outer stock diameter and inner cut-out diameter | Material in a ring-shaped piece of stock |
| Math education | R and r from a geometry problem | Verification of π(R² − r²) in geometry or calculus |
In every row, the inputs are the same: an outer radius and an inner radius. The only thing that changes is the meaning of the resulting number, which is why a calculator that returns the value in square units of whatever unit you typed — and shows the inner and outer disk areas as a cross-check — saves time across all of them.
Full Ring vs Ring Slice vs Single Circle
The formula π(R² − r²) gives the area of a complete 360° annulus — the full ring. If you only need a slice of that ring, like the wedge between two radii at a given angle, you are looking at an annulus sector rather than an annulus. The sector's area is a fraction of the full ring:
Annulus sector area = (θ / 360) × π(R² − r²) when θ is in degrees Annulus sector area = (θ / 2) × (R² − r²) when θ is in radians
An annulus sector is what you would compute for a piece of a circular path between two landmarks, or the metal area of a curved bracket. The full-ring calculator does not handle this case directly, so if your geometry problem involves an angle you will need to divide the calculator's output by 360° (or 2π radians) and multiply by the angle in degrees (or radians) you actually want.
An annulus is also easy to confuse with a single circle. A circle's area is just πr² — there is no subtraction, no inner radius, and no ring. The annulus formula only appears once a smaller circle has been removed from inside a larger one, leaving the band between them. If your shape has no inner hole, you are solving a different problem and a dedicated circle area calculator handles that case directly without any subtraction step.
Units, Validation, and Things That Go Wrong
Three small habits keep the math honest when you use the calculator or work through the formula on paper.
Keep both radii in the same unit. The calculator does not convert units for you. If you type R in centimeters and r in inches, the subtraction R² − r² mixes two unit systems and the result is meaningless. Pick one unit — meters, centimeters, millimeters, inches, feet — and use it for both inputs. The area comes back in the squared version of that unit: cm², m², in², ft², and so on.
Watch the relationship between R and r. The inner radius must be smaller than the outer radius. If r = R, the ring has no width and the area is exactly 0. If r > R, the shape simply does not exist, and the tool flags that input as invalid instead of returning a negative area — which is the right behavior, because a negative area has no geometric meaning. Negative radii are rejected for the same reason: a radius is a distance and cannot be negative. Zero and any positive length are accepted.
Pick the right tool for the right shape. If you ever need the area of a single full circle rather than a ring, that is the circle area case and uses just πr² with no inner subtraction. If you have a ring but only want a wedge of it, scale the full-ring result by the angle as described above. Mixing these cases is the most common reason people get a number that "looks right" but is actually the wrong shape.