Annulus Area Made Simple: Find the Ring Area Between Two Circles

An annulus is the flat ring you get when you punch a smaller circle out of a bigger one and keep both centered on the same point. Picture a washer, a CD, a slice through a pipe wall, or the lane on a running track. The shape is everywhere, and the moment you need to know how much surface it covers, you are doing annulus math whether you call it that or not.

The good news is that the area is not hard. You do not need calculus or any fancy integral. You need the area of two circles and one subtraction. This guide walks through the formula, runs a real example by hand, and shows where the ring area actually earns its keep.

The Formula: Outer Circle Minus Inner Circle

A full disc of radius R has area πR². That is just the standard circle formula. An annulus is that disc with a smaller disc removed, so the area is the big circle minus the small one:

annulus area = π(R² − r²)

Here R is the outer radius and r is the inner radius. Both radii are measured from the shared center. That is the whole idea: take the area you would have if the ring were filled in solid, then subtract the empty hole in the middle.

A couple of edge cases keep the formula honest. When r is 0, there is no hole and the annulus collapses into a full disc of area πR². When r equals R, the ring has zero width and the area is 0. And r must always be smaller than R — if you flip them, R² − r² goes negative and you are describing a shape that cannot exist. The annulus area calculator catches that mistake and tells you which field is which instead of printing a negative number.

A Worked Example You Can Check By Hand

Let me run one all the way through so the formula stops being abstract.

Say the outer radius is R = 10 and the inner radius is r = 6. The units do not matter for the arithmetic — centimeters, inches, meters, whatever you measured in — as long as both radii share the same unit.

Step one, square each radius:

R² = 10² = 100
r² = 6² = 36

Step two, subtract:

R² − r² = 100 − 36 = 64

Step three, multiply by π:

area = π × 64 = 64π ≈ 201.06

So the ring covers about 201.06 square units. If you measured in centimeters, that is 201.06 cm². I like leaving the answer as 64π until the last step, because it makes the arithmetic easy to audit — you can see the 64 came straight from 100 − 36, and only the final multiply needs π. When I am sanity-checking a result, I always glance at that middle number first; if R² − r² looks wrong, nothing downstream will be right.

While you are at it, two more numbers fall out for free. The ring width is R − r = 10 − 6 = 4, the straight distance from the inner edge to the outer edge. And the two circumferences are 2πR ≈ 62.83 on the outside and 2πr ≈ 37.70 on the inside.

Working From Diameters Instead of Radii

In the real world you rarely get handed a radius. A pipe is sold by its diameter, a washer is stamped with an outer size and a bore, and a tape measure gives you the distance across, not from the center out. So the most common slip is to type a diameter where the formula wants a radius.

Fix it in one move: a diameter is twice its radius, so halve every diameter before you plug it in. A pipe with a 10 cm outer diameter and an 8 cm inner diameter has R = 5 and r = 4, which gives a wall cross-section of π(25 − 16) = 9π ≈ 28.27 cm². Forgetting to halve doubles every length and, because area scales with the square, inflates your answer by a factor of four — a quiet, expensive error if you are ordering material off the back of it.

If you also need to move between units — square centimeters to square inches, say — run the result through a unit converter rather than hand-juggling conversion factors, which is its own source of slips.

Where Annulus Area Shows Up in Real Work

The formula stays the same; only the story changes. A few places I have reached for it:

Washers. A flat washer is a metal annulus, and its seating face is exactly π(R² − r²). That face area sets how the clamping load spreads under the bolt head. A washer 20 mm across the outside with an 8 mm hole has R = 10, r = 4, and a face of π(100 − 16) ≈ 263.9 mm². Compare two washer sizes and you can see which one keeps the pressure under the material limit.
Pipe walls. Slice a pipe straight across and the wall is an annulus between the outer and inner radii. That cross-section tells you how much steel the pipe contains and how it carries axial load.
Running tracks and paths. A single track lane, a circular walking path, or a donut-shaped flower bed is an annulus. Get the area and you can order gravel, turf, or soil by the square meter; the ring width R − r is how wide the lane actually is, which is the first number a contractor asks for.
Garden rings and seals. Any ring-shaped seal, gasket, or planted border is priced and sized by the same π(R² − r²).

The thread running through all of these is a round hole inside a round shape. Once you see that pattern, the annulus formula is your tool.

Solving Backwards From a Known Area

Sometimes the area is the requirement, not the result. You want a ring-shaped seal with a fixed outer radius that has to cover a specific face area, and you need to find the inner radius that delivers it. Rearrange the formula:

r = √(R² − A∕π)

For R = 5 and a target area of 16π, that gives r = √(25 − 16) = √9 = 3. There is one limit worth knowing: the most ring you can ever get from outer radius R is the full disc πR², reached when the inner radius is 0. Ask for more than that and no inner circle can leave that much ring — the math hands you the square root of a negative number, and a good tool flags it instead of inventing an answer.

If your problem involves the volume of a hollow tube rather than just its flat cross-section, the cylinder calculator takes it from there, since a hollow cylinder is an annulus extruded along its length.

Wrapping Up

The whole subject fits in one line: an annulus is a disc with a hole, and its area is the outer circle minus the inner one, π(R² − r²). Square both radii, subtract, multiply by π. Keep your radii in the same unit, halve any diameters before you start, and make sure the inner radius is the smaller of the two. Do that and the ring area is a thirty-second job — by hand or with the annulus area calculator doing the squaring and the back-solve for you.

Made by Toolora · Updated 2026-06-13