The area of an annulus is given by the formula π(R² − r²), where R is the outer radius and r is the inner radius of the ring. An annulus is the flat region between two concentric circles that share the same center, so it is the big disk minus the small disk: πR² − πr², which factors to the compact form π(R² − r²). Common examples include washers, CDs, doughnuts viewed from above, and the cross-section of a pipe wall. The Annulus Area Calculator lets you enter both radii in any unit and returns the area the moment both inputs are valid, along with the outer circle area, the inner circle area, and the ring width. The formula is airtight because of how it is built: the two circles share the same center, so subtracting one disk from the other removes only the empty hole and leaves the ring. If the inner radius is 0, the annulus collapses back to a full circle whose area is πR². If the inner radius equals the outer radius, the ring has zero width and the area is 0. That math is the entire rule, and the calculator applies it the moment you fill in two numbers.
What the Annulus Area Calculator Does
The Annulus Area Calculator takes two inputs — the outer radius R and the inner radius r — and returns the area of the ring between them. The math it runs is exactly the formula π(R² − r²); nothing more, nothing less. The output is unit-agnostic, so if you enter meters you get square meters, and if you enter centimeters you get square centimeters. Beyond the main area, the tool also reports three related values that often come in handy when you need to show your work or sanity-check the result:
| Output | Formula | What it tells you |
|---|---|---|
| Annulus area | π(R² − r²) | The area of the ring between the two circles |
| Outer circle area | πR² | The area of the full disk at radius R |
| Inner circle area | πr² | The area of the empty hole at radius r |
| Ring width | R − r | The thickness of the ring measured radially |
These four numbers are not separate calculations — they are the same formula broken down into its parts. The outer area minus the inner area always equals the annulus area, so you can use the table to verify the result by hand whenever you need to confirm the answer.
How to Calculate Annulus Area Step by Step
Running the calculation through the tool is a three-step process. You only need two numbers, and the result appears the moment both are valid.
- Type the outer radius (R) into the first box. You can enter any positive number. Use whatever unit matches your problem — meters, centimeters, inches, feet, or any other length unit. The tool does not convert units, so pick one and use it consistently for both radii.
- Type the inner radius (r) into the second box. This must be 0 or a positive number, and smaller than R. If r equals R, the ring has zero width and the area is 0. If r is larger than R, the shape does not exist and the calculator flags that with a clear message instead of returning a negative number.
- Read the annulus area right below. The radius-squared difference is multiplied by π immediately. The same display also shows the outer area, the inner area, and the ring width, so you can copy the full set of numbers into your work without re-entering anything.
That covers everything the tool needs. There are no dropdowns, no unit selectors, and no advanced settings — the formula is the formula, and the calculator just runs it for you as soon as both inputs are valid.
The Formula π(R² − r²) Explained
An annulus is the region trapped between two concentric circles — circles that share the same center but have different radii. Because they share the center, every point inside the bigger circle is also inside the annulus, except for the points that are inside the smaller circle. That makes the area a straightforward subtraction:
Area of annulus = Area of outer circle − Area of inner circle = πR² − πr²
Since π is a common factor, you can pull it out front and combine the two squared terms:
= π(R² − r²)
Both forms are correct. The expanded form πR² − πr² is useful when you want to compute each disk's area separately (for example, in a report where you need to show both circles). The factored form π(R² − r²) is what most textbooks and the calculator prefer because it requires only one subtraction inside the parentheses before multiplying by π. The two expressions are mathematically identical, so they always give the same numeric answer.
The factored form is also faster to compute by hand if you already know R² − r². If R = 10 and r = 6, the difference of squares 100 − 36 = 64, so the area is 64π — no need to subtract two decimal π multiples separately.
A Worked Example with the Calculator
To make the math concrete, take an annulus with outer radius R = 10 and inner radius r = 6, both in the same unit. Plugging into the formula step by step:
Outer circle area = π × 10² = 100π ≈ 314.16 square units Inner circle area = π × 6² = 36π ≈ 113.10 square units Annulus area = π(10² − 6²) = π(100 − 36) = 64π ≈ 201.06 square units Ring width = 10 − 6 = 4 units
Each output in the table above maps to one of these numbers. The annulus area of 64π can be verified by subtracting the inner area from the outer: 100π − 36π = 64π. The calculator shows all four values at once, so you can use the same example to teach the formula or to double-check a homework problem.
Units and Edge Cases the Tool Handles
The Annulus Area Calculator is unit-agnostic, which means it does not carry any unit information inside the math. Whatever unit you choose for the radii, the area comes out squared. If you enter R = 0.3 and r = 0.1 in meters, the area is 0.08π ≈ 0.251 square meters. If you enter the same numbers in millimeters, the area is 0.08π ≈ 0.251 square millimeters. The number is the same; the unit label is whatever you started with.
Three edge cases are worth knowing about, all of which the calculator handles cleanly:
- r = 0: The annulus collapses to a full disk of radius R. The formula gives πR², which is the area of a single circle. The inner area output reads 0, which is correct.
- r = R: The ring has zero width, so its area is exactly 0. The width output reads R − R = 0 and the area reads 0. The math is satisfied without any special case.
- r > R: The shape is no longer an annulus, so a negative area would be meaningless. The calculator flags this with a clear message instead of printing a negative number.
Negative radii are rejected because a radius is a distance — by definition it cannot be negative. So the input contract is simple: non-negative numbers, with r smaller than R. Everything runs locally in your browser, so nothing is uploaded and the result updates the moment you finish typing.
Where Annulus Area Shows Up in Real Work
The annulus shape is more common than it looks at first glance. Engineers reach for π(R² − r²) when they compute the load-bearing cross-section of a hollow shaft, a tube, or a pipe wall — the metal that actually carries the load is the ring between the outer and inner diameters. The same formula gives the metal area in a washer, a flange, or a round gasket, which is why machine parts are often priced by mass and material density.
Architects and landscapers use it for circular paths, pond rims, and individual running-track lanes, where each lane is an annulus between two radii and the area of the rubber or paving tells you how much material to order. Machinists use it for ring-shaped stock where the waste material in the middle has to be subtracted from the outer disk. In food service, an annulus is the right shape for a pizza crust, a doughnut, or the edible band of a ring-shaped cake.
Students meet the formula in geometry, where it is usually followed by the question of how to compute the area of a sector of an annulus — a slice of the ring bounded by two radii instead of a full 360°. For that case, the area is (θ/360) × π(R² − r²) in degrees, or (θ/2) × (R² − r²) in radians. The standard Annulus Area Calculator computes the full ring; for a slice, the area scales linearly with the angle.
From sizing a pipe to packing a gasket, from ordering paving for a circular path to finishing a geometry problem, the math is the same: π(R² − r²). Open the Annulus Area Calculator, type the two radii, and the answer — plus the outer area, inner area, and ring width — is in front of you.