Annulus Area Calculator – Find the Area of a Ring Shape
An annulus is the flat, ring-shaped region between two concentric circles — circles that share the same centre point but have different radii. It is the area you get when you remove a smaller disc from a larger disc. Everyday examples include washers, pipe cross-sections, circular frames, and the rings of a target board.
What Is an Annulus?
The outer boundary of an annulus is the larger circle with radius R(the outer radius), and the inner boundary is the smaller circle with radius r (the inner radius). The ring width w = R − r is the radial thickness of the annular region. Both radii must be positive, and the inner radius must be strictly less than the outer radius for a valid annulus to exist.
Annulus Area Formula
The area of an annulus is calculated by subtracting the inner circle area from the outer circle area:
Area = π × R² − π × r² = π × (R² − r²)
An equivalent and sometimes more convenient form uses the ring width w = R − r:
Area = π × (R + r) × (R − r) = π × (R + r) × w
Where π ≈ 3.14159265. Both forms give identical results; the second is useful when the ring width is known directly.
Step-by-Step Example
Suppose a steel washer has an outer radius of 10 cm and an inner radius of 6 cm:
- Outer circle area = π × 10² = 314.1593 cm²
- Inner circle area = π × 6² = 113.0973 cm²
- Annulus area = 314.1593 − 113.0973 ≈ 201.062 cm²
- Ring width = 10 − 6 = 4 cm
Using the alternative formula: Area = π × (10 + 6) × 4 = π × 64 ≈ 201.062 cm² — exactly the same answer.
Unit Support and Conversion
This calculator accepts radii in seven length units: millimetres (mm), centimetres (cm), metres (m), kilometres (km), inches (in), feet (ft), and yards (yd). Both radii must be in the same unit. The result is expressed in the corresponding square unit (e.g., cm² when the unit is cm). The multi-unit table converts the same annulus area into all seven area units simultaneously — useful when your output must match a specific unit required by a drawing or specification.
Width-Based Entry
In many practical situations — such as measuring a machined ring or a gasket — the ring width is easier to measure directly than the inner radius. Switch to Width Mode and enter the inner radius and ring width. The calculator derives the outer radius as R = r + w and computes all outputs as normal. This avoids the need to subtract manually before entering values.
Practical Applications
Annulus area calculations arise in many fields:
- Mechanical engineering — computing the cross-sectional area of hollow shafts, tubes, or pipe walls to determine material volume and weight.
- Civil engineering — calculating the ring area of circular concrete columns, tunnels with circular linings, or drainage pipes.
- Manufacturing — determining the area of washers, O-rings, or disc-shaped gaskets for material cost estimates and press-fit calculations.
- Architecture and design — finding the area of circular frames, windows with circular panes, or the ring sections of curved facade panels.
- Fluid mechanics — computing the flow area of an annular duct (e.g., the space between a pipe and a concentric inner tube) for flow rate and velocity calculations.
- Astronomy — estimating the apparent ring area of a solar annular eclipse or planetary ring system cross-section when viewed edge-on.
Annulus vs Circle vs Sector vs Segment
It helps to distinguish between related circular shapes:
- Circle area — the full area enclosed by a single circle:
πr². - Annulus area — the ring between two concentric circles:
π(R² − r²). This tool. - Sector area — the "pie slice" bounded by two radii and an arc.
- Segment area — the region between a chord and the arc it subtends.
The annulus is unique because it involves two separate circles, whereas sectors and segments are sub-regions of a single circle.
Tips for Accurate Results
Use consistent units for both radii — mixing centimetres and millimetres will produce an incorrect result. Increase the decimal precision setting if your application requires more than four significant figures. All intermediate computations use full floating-point precision; rounding only occurs at the display stage. The formula is numerically stable for any valid pair of radii regardless of their magnitude.