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Annulus Area Calculator

Instantly find the area of a ring (annulus) with π(R² − r²).

Privacy: your files never leave your device. All processing happens locally in your browser.

How to use

  1. 1.Type the outer radius (R) into the first box, in any unit you like.
  2. 2.Type the inner radius (r) into the second box — it must be smaller than R.
  3. 3.Read the annulus area instantly below; it also shows the outer area, inner area, and ring width.

About Annulus Area Calculator

The area of an annulus is π(R² − r²), where R is the outer radius and r is the inner radius. This annulus area calculator returns that value the instant you enter both radii, and it also shows the outer circle area (πR²), the inner circle area (πr²), and the ring width (R − r) so you can check every part of the result.

An annulus — often just called a ring — is the flat region between two concentric circles that share the same center. Think of a washer, a CD, a doughnut seen from above, or the cross-section of a pipe wall. Because the two circles are concentric, the ring's area is simply the big disk minus the small disk: πR² − πr², which factors to the compact form π(R² − r²). That single subtraction is the whole idea: you remove the empty hole in the middle from the full outer circle.

The inner radius must be smaller than the outer radius. If r equals R the ring collapses to a line and its area is exactly 0; if r were larger than R the shape would not exist, so a negative "area" is meaningless. This calculator blocks those cases with a clear message instead of printing a negative number. A radius of 0 or any positive length is accepted, while negative radii are rejected because a radius is a distance.

Where does this matter? Engineers use it for the load-bearing cross-section of hollow shafts, tubes and pipes, for the metal area of a washer or flange, and for the material in a round gasket. Architects and landscapers use it for circular paths, pond rims and running-track lanes, where each lane is an annulus between two radii. Machinists compute it for ring-shaped stock, and students meet it throughout geometry and calculus.

Use any unit consistently: if you enter the radii in centimeters, the area comes out in square centimeters; enter meters and you get square meters. The tool never converts units for you — it keeps the math pure so the numbers are exactly π(R² − r²).

A quick worked example: with R = 10 and r = 6, the outer area is 100π ≈ 314.16, the inner area is 36π ≈ 113.10, and the annulus area is (100 − 36)π = 64π ≈ 201.06 square units. Everything runs locally in your browser — nothing is uploaded — so results are instant and private.

Frequently asked questions

What is the formula for the area of an annulus?
The area of an annulus is π(R² − r²), where R is the outer radius and r is the inner radius. It equals the outer circle's area minus the inner circle's area. For example, with R = 10 and r = 6 the area is 64π ≈ 201.06 square units.
Why must the inner radius be smaller than the outer radius?
An annulus is the gap between two concentric circles, so the inner circle has to sit inside the outer one. If r is larger than R the ring cannot exist and the formula would give a negative number, so this calculator flags that as invalid rather than showing a negative area. If r equals R the ring has zero width and the area is 0.
What is the difference between an annulus and an annulus sector?
An annulus is the full ring between two concentric circles — a complete 360° band. An annulus sector is only a slice of that ring spanning an angle θ, so its area is a fraction: (θ/360) × π(R² − r²) in degrees, or (θ/2) × (R² − r²) in radians. This tool computes the full annulus.
What units does the annulus area calculator use?
It is unit-agnostic. Enter both radii in the same unit and the area comes out squared: centimeters give square centimeters, meters give square meters, inches give square inches. The tool does not convert units, so keep R and r in one consistent unit.
How do I find the width of the annulus?
The ring width is simply the outer radius minus the inner radius, R − r. The calculator shows this next to the area. For instance, R = 10 and r = 6 gives a width of 4 units. Width is a length, while the annulus area — π(R² − r²) — is in square units.

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