Annulus Calculator
Enter the outer and inner radii of an annulus — the flat ring between two concentric circles — for its area, both circumferences and the ring width, drawn to scale. You can also click a value on the drawing to change it. For the 3D version, a ring with length, see the pipe calculator.
Dimensions
R > rOuter radius and inner radius — everything else follows.
Results
Annulus formulas
An annulus is the region between two concentric circles. Its area is simply the big circle minus the hole, and the width of the ring is the difference of the radii.
Area · A = π (R² − r²)
Width · w = R − r
Circumferences · 2πR and 2πr
A useful factorisation
Because R² − r² = (R + r)(R − r), the area is also the mean circumference times the width: A = π(R + r) × w. A running track's area is its centreline length times its lane width.
Tips
Pick your units in the Display row — areas are shown in square units automatically. The inner radius must be smaller than the outer; the drawing keeps both true to scale.
How to find the area of an annulus
An annulus is the flat ring between two concentric circles — a washer, a gasket, the face of a pipe, a running track, the crust of a pizza slice's parent pie. Two measurements fix it completely: the outer radius R and the inner radius r. The area is the outer circle's area with the hole subtracted.
A = πR² − πr² = π(R² − r²)
The difference of squares factors as (R + r)(R − r), which gives the same area a second, very practical reading: the mean circumference, π(R + r), multiplied by the ring width, R − r. If you unrolled the ring into a (nearly) straight strip, that's its length times its width.
Worked example
Take a washer with an outer radius of 12 and an inner radius of 5.
Square both radii and take the difference.
R² − r² = 144 − 25 = 119
The ring area follows directly.
A = 119π ≈ 373.85
The width is the difference of the radii; each edge is an ordinary circle.
w = 12 − 5 = 7 · outer C = 24π ≈ 75.40 · inner C = 10π ≈ 31.42
Common mistakes
The classic error is subtracting the radii before squaring: π(R − r)² is not the ring area — for the washer above it would give 49π instead of 119π. Square first, then subtract. The other frequent slip is mixing a radius with a diameter; if you have diameters, halve them both before starting.
Where it comes up
Annulus area is the cross-section behind pipe and tube weight calculations — multiply it by the length and density and you have the mass, which is exactly what the pipe volume calculator does. It also covers washers and gaskets, the material in a CD or vinyl record, circular walkways and garden borders around ponds, and the area of a running track between its kerbs.
Frequently asked questions
How do you find the area of an annulus?
Square both radii, subtract, and multiply by π: A = π(R² − r²). With an outer radius of 12 and inner radius of 5, that's π(144 − 25) = 119π ≈ 373.85.
Why can't I just use π(R − r)²?
Because the square of a difference is not the difference of squares. π(R − r)² is the area of a small circle whose radius equals the ring width — a different shape entirely. Square first, then subtract.
What is the width of an annulus?
The distance across the ring, w = R − r. Handily, the area is also the mean circumference times this width: A = π(R + r) × w.
I have diameters instead of radii — what do I do?
Halve them both: R = D⁄2 and r = d⁄2. In one step the area is A = π(D² − d²)⁄4.
How is an annulus related to a pipe?
A pipe is an annulus extruded along a length: its cross-section is exactly this ring, and its wall volume is the ring area times the length, V = π(R² − r²)h. The pipe calculator handles the 3D version.