An annulus is the region between two concentric circles (circles that share the same centre). It looks like a flat ring or washer. The annulus is defined by its outer radius R and inner radius r, where R must be greater than r. It appears in engineering for cross-sections of hollow pipes and tubes, in optics for lens apertures, and in statistics for confidence bands around a circle.

What is the area formula for an annulus?

The area of an annulus equals the area of the outer circle minus the area of the inner circle: pi times R squared minus pi times r squared, which simplifies to pi times (R squared minus r squared), or equivalently pi times (R + r) times (R - r). This last form is useful when you know the sum and difference of the radii rather than the individual values.

What is the width of an annulus?

The width (or thickness) of an annulus is simply the difference between the outer and inner radii: width = R minus r. It tells you how wide the ring is at any point, since the gap between the two circles is constant all the way around.

Annulus Calculator

An annulus is the flat ring-shaped region between two concentric circles. Concentric means both circles share the same centre point but have different radii. The outer circle defines the outer boundary and the inner circle defines the hole in the middle, making the annulus look like a washer or a donut cross-section. Annuli appear throughout engineering in the cross-sections of hollow cylinders such as pipes, tubes, and gun barrels, where knowing the ring area is essential for calculating flow rate, stress, or material volume per unit length. They also appear in optics when describing annular apertures, in astronomy for ring-shaped nebulae, and in geometry problems involving concentric circular regions. This calculator takes the outer radius R and the inner radius r and returns the annulus area, the outer circumference, the inner circumference, the ring width (the difference between the two radii), and the equivalent solid circle radius that would have the same area. The inner radius must be smaller than the outer radius; the area is zero only if they are equal. The default uses outer r = 8 and inner r = 5, giving area = 122.52, outer C = 50.27, and inner C = 31.42.

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Outer radius (R)

units

Inner radius (r)

units

122.52

annulus area (square units)

Outer circumference50.27

Inner circumference31.42

Ring width3.00

Equivalent radius6.24

Inner radius must be less than outer radius. Equivalent radius is the radius of a solid circle with the same area.

How it works

For outer radius R and inner radius r: area = pi * (R^2 - r^2). Outer circumference = 2 * pi * R. Inner circumference = 2 * pi * r. Ring width = R - r. Equivalent radius = sqrt(R^2 - r^2), derived by setting pi * R_eq^2 equal to the annulus area and solving for R_eq. All four measurements update simultaneously when you change either radius.

Worked example

For outer r = 8 and inner r = 5: area = pi * (64 - 25) = pi * 39 = 122.52. Outer circumference = 2 * pi * 8 = 50.27. Inner circumference = 2 * pi * 5 = 31.42. Ring width = 8 - 5 = 3.00. Equivalent radius = sqrt(39) = 6.24. These match the defaults pre-filled above.

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