Annulus Area Calculator
From an outer and an inner radius, get the area of the ring and how wide it is — the two numbers that describe any flat ring.
Big circle minus the hole
Enter the outer radius R and the inner radius r, and the calculator returns the ring area π(R² − r²) and the ring width R − r at once.
Inner must be smaller
The inner radius has to be smaller than the outer one — otherwise the hole swallows the disc and there is no ring to measure.
What is an annulus?
Two radii in, one flat ring out
An annulus is a flat ring — the region trapped between two concentric circles that share the same centre. The outer circle has radius R, the inner hole has radius r, and the annulus is everything in between. You meet it everywhere round things have holes: washers, the cross-section of a pipe wall, a CD or vinyl record, the lane area of a running track, a doughnut seen from above. Once you know both radii, two numbers describe the ring completely — its area (the surface it covers) and its width (how thick the band is) — and this calculator returns both at once.
Enter the outer and inner radii in any length unit to get the ring area and width instantly.
The area of an annulus is just the area of the big circle minus the area of the hole, and the width is the difference between the two radii.
area = π × (R² − r²)Because the area of a full circle is π × radius², the outer disc has area π × R² and the hole has area π × r². Subtract one from the other and factor out π to get π × (R² − r²). The width — the radial thickness of the band — is simply R − r and does not involve π at all.
Suppose you have a ring with an outer radius of 5 and an inner radius of 3.
Square each radius
R² = 25 and r² = 9 — the areas of the two circles divided by π.
Subtract and multiply by π
π × (25 − 9) = π × 16 = 50.265482 square units — the ring area.
Width
5 − 3 = 2 — how thick the band is from inner edge to outer edge.
The two outputs answer two different everyday questions. The ring area (about 50.265482 square units for R = 5 and r = 3) is the surface the ring actually covers — the metal in a washer, the material in a pipe wall's cross-section, the paint on a circular border. It is the big circle's area minus the hole, so a small change in the outer radius matters far more than the same change in the inner radius, because area grows with the radius squared. The ring width (2 in this example) is the plain radial thickness — how far the band stretches from the inner edge to the outer edge along a straight line through the centre. Notice that width and area are independent ideas: two rings can share the same width of 2 yet have very different areas if one sits near the centre and the other far out. The one rule that always holds is that the inner radius must be smaller than the outer radius; if they are equal there is no ring at all, and if the inner one is larger the shape is meaningless.
The formula is exact, but a couple of practical points are worth keeping in mind.
Perfect rings and consistent units
This formula describes a perfect, flat annulus with two truly concentric circles. Real objects — a slightly off-centre hole, a pipe with a rough bore, a washer with chamfered edges — will differ a little from the computed value. Both radii are also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: radii in centimetres give a width in centimetres and an area in square centimetres, never a mix.