Annulus: Area, Circumference, Width, and Formula

An annulus is the region between two circles that have the same center. In everyday language, it looks like a flat ring. A washer, the cross-section of a hollow pipe, and the space between two circular boundaries can all have the geometry of an annulus.

The useful idea is to stop thinking of the annulus as a completely new shape. It is simply a large circle with a smaller circular region removed from its center. Once that is clear, its area and circumference formulas follow directly from the formulas for a circle.

Annulus Area Formula: The Basic Idea

The area of a circle with radius R is

If a smaller circle of radius r is removed from the middle, its area is

Therefore, the area of the annulus is the larger circle area minus the smaller circle area:

Factoring out π gives the usual compact form:

This is the main formula to remember. The outer radius must be larger than the inner radius, so for an ordinary annulus, R > r.

Why the formula makes geometric sense

Imagine filling the entire outer circle with paint. Now remove everything inside the smaller circle. The paint that remains is exactly the annulus.

So there is no separate mystery behind the area formula. It is the ordinary circle formula applied twice and then subtracted.

A simple example

Suppose an annulus has an outer radius of 12 cm and an inner radius of 5 cm. Substitute the two radii into the formula.

The important step is the subtraction of the squared radii. It is not correct to calculate π(12 − 5)². The radii are squared first and then subtracted.

Annulus Circumference: Outer and Inner Circle Formulas

An annulus has two circular boundaries. The outside boundary has the circumference of a circle with radius R:

The inside boundary has the circumference of a circle with radius r:

If a problem asks for the total length around both boundaries, add them:

This distinction matters. The area of the annulus describes the material in the ring. The total boundary length describes both circular edges.

When diameters are given instead of radii

Many practical measurements are given as diameters. A pipe, washer, or circular opening may be specified by its outside diameter and inside diameter rather than by its radii.

If D is the outer diameter and d is the inner diameter, then

Substitute these into the annulus area formula:

which simplifies to

For example, if a metal washer has an outside diameter of 100 mm and an inside diameter of 80 mm:

The unit is square millimeters because the calculation describes an area.

Ring width or thickness

The radial width of an annulus is the difference between the two radii:

Notice that this is a length, not an area. If R = 12 cm and r = 5 cm, the ring is 7 cm wide in the radial direction.

A useful form of the area formula

The difference of two squares can be factored:

Since R − r is the ring width w, the area can also be written as

This version is surprisingly useful. It says that the area of the ring is related to its width and to the average size of its two circular boundaries.

For a thin ring, this gives a good geometric way to think about the calculation. If the inner and outer radii are close together, the annulus resembles a long circular strip that has been wrapped around the center.

Finding a missing radius from the area

Sometimes the area of the annulus and one radius are known, while the other radius must be found.

Starting from

if R and A are known and r is unknown, first divide by π:

Then rearrange:

and therefore

For example, suppose the outer radius is 10 cm and the annulus area is 50π cm².

The same process can be reversed if the inner radius and annulus area are known and the outer radius is required:

Annulus geometry in a pipe cross-section

A hollow cylindrical pipe provides a useful real-world example. If you look directly at the end of the pipe, the metal cross-section is an annulus.

If the pipe has length L, the volume of the material can be found by multiplying the annulus area by the length:

This is the same idea used for a cylinder: cross-sectional area multiplied by length. The only difference is that the cross-section is a ring rather than a solid disk.

What Happens When the Inner Radius Is Zero?

If r = 0, there is no hole in the middle. The annulus becomes an ordinary circle.

So a solid disk can be viewed as the limiting case of an annulus in which the inner circle has shrunk to a single point.

Common Annulus Calculation Mistakes to Avoid

The first common mistake is confusing radius and diameter. If the outside diameter is 20 cm, the outer radius is 10 cm, not 20 cm.

The second is subtracting the radii before squaring them. The correct expression is

not

These expressions are generally different.

Another mistake is forgetting that the two circles must have the same center for the usual annulus formulas to apply. The ring is defined by two concentric circles.

Finally, keep track of units. Radii and circumferences are measured in units such as cm or inches, while area is measured in square units such as cm² or in². If a length is later used to calculate the volume of a cylindrical object, the result becomes a cubic unit.

Annulus Formula Summary: The Key Ideas

An annulus is best understood as a circle with a circular hole removed from its center. From that single idea, the principal formulas follow naturally.

If diameters are given, convert them to radii or use the equivalent area form:

The most useful habit is to identify what each measurement represents before doing any arithmetic. Once the outer boundary, inner boundary, and their common center are clear, annulus problems are usually just ordinary circle geometry applied carefully.