Annulus (Ring) Area Calculator
Enter the outer and inner radii of a flat circular ring and the calculator returns the annulus area A = π(R² − r²), along with the ring width and the two bounding circumferences.
Annulus area (A = π(R² − r²))
201.06193 square units
- Outer radius (R)
- 10
- Inner radius (r)
- 6
- Ring width (R − r)
- 4
- Outer circle area (πR²)
- 314.16
- Inner circle area (πr²)
- 113.1
- Outer circumference (2πR)
- 62.83
- Inner circumference (2πr)
- 37.7
An annulus is the flat ring between two concentric circles. Its area is the outer disc area minus the inner disc area: A = πR² − πr² = π(R² − r²) = π(R + r)(R − r).
How to use this calculator
Enter the outer radius (R) of the ring and the inner radius (r) of the hole in the middle. Both must be in the same length unit — centimetres, metres, inches, feet, whichever you prefer. The headline result is the area of the ring in those squared units, and the breakdown shows the ring width (R − r), each disc area, and each circumference. The inner radius must be non-negative and strictly smaller than the outer radius; setting r = 0 collapses the annulus to a full circle of radius R.
How the calculation works
An annulus is the flat region between two concentric circles. Its area is the area of the outer disc minus the area of the inner disc: A = πR² − πr², which factors as A = π(R² − r²) = π(R + r)(R − r). The first form is easier to compute when you already have both radii; the factored form makes it obvious that the area grows linearly in the ring width (R − r) once the mean radius (R + r)/2 is fixed. The two circumferences are the standard 2πR and 2πr.
Worked example
Take an outer radius of 10 cm and an inner radius of 6 cm. The outer disc has area π·10² = 100π ≈ 314.159 cm² and the inner disc has area π·6² = 36π ≈ 113.097 cm². The annulus area is the difference: 100π − 36π = 64π ≈ 201.062 cm². Equivalently, π(10 + 6)(10 − 6) = π·16·4 = 64π — the factored form gives the same answer. The ring width is 10 − 6 = 4 cm.
Frequently asked questions
What is an annulus?
An annulus is the flat ring-shaped region lying between two circles that share the same centre. It is bounded on the outside by a circle of radius R and on the inside by a smaller circle of radius r. Real-world examples include a washer, the cross-section of a hollow pipe, a flat doughnut (the donut shape, but in two dimensions), a CD or vinyl record, the rim of a lens, and the orbital "ring" around a star. In Latin, annulus simply means "little ring".
What is the formula for the area of an annulus?
A = π(R² − r²), where R is the outer radius and r is the inner radius. The formula is just the area of the outer circle πR² minus the area of the hole πr². It can also be written A = π(R + r)(R − r) using the difference-of-squares identity — the factored form is sometimes more convenient when you already know the ring width (R − r) and the sum (R + r). Both forms give the same number; pick whichever has the easier arithmetic.
Can I use diameter instead of radius?
You can, but you have to convert first: the radius is always half the diameter. If you have the outer and inner diameters D and d, the annulus area is π((D/2)² − (d/2)²) = (π/4)(D² − d²). The calculator on this page takes radii; halve your diameter values before entering them. The formula in terms of diameter is the one you will see most often for pipes and tubes, where diameter is the natural specification.
How is this different from a torus?
An annulus is a two-dimensional flat ring — a region of the plane bounded by two concentric circles. A torus is the three-dimensional doughnut shape you get when you spin a circle around an axis that does not intersect it. The annulus has an area; the torus has a surface area and a volume. The two shapes are related (the torus is generated by rotating a small disc whose cross-section is bounded by an annulus), but they are not the same object. This calculator handles only the 2D annulus.
What is the ring width and why does the calculator report it?
The ring width is simply R − r, the radial distance from the inner edge to the outer edge of the annulus. It is the thickness of the ring measured along any radius. The ring width often matters more than the area in practical settings: a washer is specified by its inner and outer diameters and so implicitly by its width; a pipe wall has a thickness that is exactly the ring width; an athletics track lane has a width. We report it alongside the area because the two together fully describe the ring.
What if the outer and inner radii are equal?
Then the annulus has zero area — it has collapsed to a single circle (a curve), not a region. The calculator rejects this case (and any case where the inner radius is bigger than the outer) and asks you to re-enter the inputs. If you genuinely have R = r — say, an infinitely thin ring — the limiting area is zero. If r is just slightly less than R, the annulus area is approximately 2πR · (R − r), the outer circumference times the small width; that is the first-order approximation used in physics for "thin shells".