Calculator Features
🧭 Jump to a section
- What is an annulus
- Annulus formula
- Why this formula works
- Ring width
- Inner vs outer radius
- Area between two circles
- Pipe cross-sections
- Washers & gaskets
- Engineering standards examples
- Worked examples
- Engineering applications
- Manufacturing applications
- Construction applications
- Glossary
- Common mistakes
- Common materials & densities
- Practice questions
- Printable formula sheet
What Is an Annulus?
An annulus — also called a ring, circular ring, or annular ring — is the flat region between two circles that share the same centre (concentric circles), one inside the other. It's essentially a hollow circle: a full disc with a smaller circular hole removed from its centre. Every annulus is fully determined by just two independent measurements: the outer radius (R) and the inner radius (r), with R always greater than r.
A labeled annulus: outer radius, inner radius, ring width, and the shaded ring area itself.
Annulus Formula
The area between the two circles is simply the outer circle's area minus the inner circle's area. A ring with outer radius 10 and inner radius 6 has an area of π×(100−36) = π×64 ≈ 201.062 square units.
Why This Formula Works
An annulus's area is defined as Outer Circle Area − Inner Circle Area, and the reasoning is genuinely simple once you see it: the outer circle (area πR²) fully covers the annulus and the hole in the middle. Removing exactly the hole — the inner circle, area πr² — leaves only the ring. There's no approximation or special geometry involved, just straightforward subtraction:
This is why an annulus, unlike a sector or segment, never needs trigonometry — it's built entirely from two ordinary circle-area calculations.
Ring Width
Ring width (also called thickness) is a completely different measurement from area, and it's easy to mix the two up. Width is a simple linear distance — how far it is from the inner edge to the outer edge, measured along a radius. Area is a two-dimensional quantity. Two rings can have the same width but very different areas if their radii differ, and vice versa.
Inner vs Outer Radius
The outer radius (R) is the distance from the centre to the ring's outer edge; the inner radius (r) is the distance from the centre to the ring's inner edge (the edge of the hole). R must always be strictly greater than r — if they were equal, there would be no ring left at all, and this calculator rejects that combination with a clear explanation.
Area Between Two Circles
"Area between two circles" is simply another way of describing annulus area, especially in search queries and general usage. As long as the two circles are concentric (share a centre), the calculation is identical: subtract the smaller circle's area from the larger one's, A = π(R²−r²). If the two circles are not concentric, the overlapping region is a more complex shape (a lens or crescent), and this calculator's formulas do not apply.
Pipe Cross-Sections
A pipe's cross-section — the flat end-view of the tube wall — is exactly an annulus. Pipes are usually specified by outer diameter (OD) and wall thickness (t) rather than by inner and outer radius directly, which is why this calculator includes a dedicated Pipe Dimensions mode: inner radius = R − t, where R = OD/2.
Washers & Gaskets
Flat washers, O-rings (in their flattened cross-sectional sense), and gaskets are all real-world annuli. Their area determines how much load-bearing surface a washer provides, or how much sealing surface a gasket covers — both directly calculated from outer and inner diameter using the same annulus formula.
Engineering Standards Examples
Real standardised dimensions, so you can check these figures against the manufacturer's own spec sheet:
ISO 7089 M10 flat washer — inner ⌀10.5 mm, outer ⌀21 mm
DN50 (2") pipe — outer diameter 60.3 mm (the internationally standard OD for this nominal size), wall thickness 3.91 mm (Schedule 40 reference)
ANSI flanges are another common real-world annulus — the bolt-hole face of a flange is a ring — but flange face dimensions vary meaningfully by pressure class (150#, 300#, and so on), so no single "ANSI flange" number applies universally; check the specific class's dimensional table for exact figures rather than a generic worked example.
Worked Examples
M8 washer — outer radius 8 mm, inner radius 4.2 mm
DN100 pipe — outer diameter 114.3 mm, wall thickness 6 mm
Rubber gasket — outer radius 30 mm, inner radius 20 mm
Machine spacer — outer radius 50 mm, inner radius 35 mm
Radius example — outer radius 25, ring width 5 (solve inner radius)
Diameter example — CD/disc, outer diameter 120 mm, inner diameter 15 mm
Material volume & weight — steel spacer, R=50 mm, r=35 mm, depth 20 mm
Impossible-input example
Engineering Applications
| Field | Use |
|---|---|
| Mechanical engineering | Bearing races, bushings, and rotating ring components where cross-sectional area affects strength and stiffness |
| Fluid engineering | Pipe wall cross-sectional area for flow calculations, pressure ratings, and material volume |
| Structural engineering | Hollow circular columns and tubular structural members, where the annulus determines section properties |
| Aerospace | Turbine and engine components with concentric ring cross-sections |
Manufacturing Applications
| Use | How annulus geometry helps |
|---|---|
| Washer and gasket production | Material quantity and weight per unit are calculated directly from annulus area and material density |
| CNC ring cutting | Toolpaths for ring-shaped parts are programmed using outer and inner radius directly |
| Sheet metal stamping | Blank size and scrap (the inner disc removed) are both annulus-related calculations |
| Quality control | Measured width and area are cross-checked against design tolerances |
Construction Applications
| Use | How annulus geometry helps |
|---|---|
| Circular footings and foundations | Ring-shaped foundation pads (e.g. around a central column) use annulus area for concrete volume estimates |
| Manhole and access covers | The load-bearing ring around a circular opening is an annulus |
| Circular tank walls | Wall cross-sectional area (plan view) for material estimation |
Glossary
- Annulus
- The region between two concentric circles; also called a ring.
- Concentric circles
- Two or more circles that share exactly the same centre point.
- Outer radius (R)
- The distance from the shared centre to the annulus's outer edge.
- Inner radius (r)
- The distance from the shared centre to the annulus's inner edge (the edge of the hole).
- Ring width (thickness)
- The distance from the inner edge to the outer edge, w = R − r.
- Mean radius
- The average of the outer and inner radius, (R+r)/2 — useful for approximating thin-ring calculations.
- Area ratio
- The annulus area as a fraction of the full outer circle's area.
- Wall thickness
- The pipe-industry term for ring width, usually denoted t.
Common Mistakes
| Mistake | Fix |
|---|---|
| Confusing ring width with area | Width (R−r) is a linear distance; area (π(R²−r²)) is two-dimensional — they scale completely differently |
| Using diameters directly in the area formula | A = π(R²−r²) needs radii — halve any diameters first, or use this calculator's diameter-based modes directly |
| Assuming area scales linearly with width | Two rings of the same width can have very different areas depending on their radii — area depends on R²−r², not just R−r |
| Entering inner radius larger than outer radius | The outer radius must always be the larger of the two — swap them if you've mixed them up |
| Forgetting to convert wall thickness correctly for pipes | Wall thickness in pipe specs is a radius-direction measurement (R−r), not related to diameter directly without halving first |
Common Materials & Densities
Reference densities for the optional weight calculator above — select any of these directly from the material dropdown, or type your own value if your material isn't listed:
| Material | Density (g/cm³) |
|---|---|
| Steel (mild) | 7.85 |
| Aluminium | 2.70 |
| Brass | 8.50 |
| Copper | 8.96 |
| Stainless Steel (304) | 7.93 |
| Cast Iron | 7.20 |
| PVC | 1.40 |
| Rubber | 1.10 |
Densities are typical room-temperature values for general reference; alloy grade, temperature and manufacturing process can shift real material density slightly — check the specific mill certificate or datasheet for critical engineering work.
Formula Cheat Sheet
Quick Reference
Area:A = π(R²−r²) | Width: w = R−rOuter circumference:
Co = 2πR | Inner circumference: Ci = 2πrMean radius:
Rm = (R+r)/2 | Mean diameter: Dm = R+rOuter radius from area+inner:
R = √(r²+A/π) | Inner radius from area+outer: r = √(R²−A/π)Pipe inner radius:
r = OD/2 − t
Practice Questions
Beginner (with answers)
- Find the area of an annulus with outer radius 8 and inner radius 5.
- Find the ring width for outer radius 12 and inner radius 7.
- An annulus has outer diameter 20 and inner diameter 10. Find its area.
- Can the inner radius equal the outer radius?
- Find the mean radius for R=10, r=6.
Show answers
1) π×(64−25)≈122.522 2) 12−7=5 3) R=10,r=5, π×(100−25)≈235.619 4) No — that would leave zero ring width 5) (10+6)/2=8
Advanced (with answers)
- An annulus has area 100 and inner radius 4. Find the outer radius.
- An annulus has area 150 and outer radius 10. Find the inner radius.
- A pipe has OD 50 mm and wall thickness 5 mm. Find the inner diameter.
- Find the area ratio (annulus area ÷ outer circle area) for R=10, r=8.
- An annulus has outer circumference 62.83 and inner radius 3. Find its area.
Show answers
1) R=√(16+100/π)≈6.92 2) r=√(100−150/π)≈7.23 3) R=25,r=20, inner diameter=40mm 4) (100−64)/100=0.36 5) R=62.83/2π=10, area=π×(100−9)≈285.88
🔑 Key Takeaways
- An annulus is the region between two concentric circles: area A = π(R²−r²), width w = R−r.
- Width and area measure completely different things — never assume one scales with the other.
- An annulus has no trigonometry involved at all, unlike sectors, segments, or chords.
- Pipes are conventionally specified by outer diameter and wall thickness, not radii directly.
- This calculator's eight modes cover every practical starting combination, from raw radii to pipe-industry OD+wall-thickness conventions.
Frequently Asked Questions
Calculation Assumptions
ℹ️ What this calculator assumes
- Results are calculated using Euclidean (flat-plane) geometry.
- The two circles are assumed to be concentric (sharing exactly the same centre) throughout every mode.
- Rounding to the selected decimal precision may cause very small differences between displayed values and hand calculations.
- All input values are assumed to use the same unit — mixing units (e.g. an outer radius in metres and an inner radius in centimetres) will produce an incorrect result.
- All eight solve modes have exact closed-form algebraic solutions — no numerical approximation or trigonometry is used anywhere on this page.
- Material volume and weight are optional extras, calculated only when a depth and/or density are entered; density is assumed in grams per cubic centimetre (g/cm³).
About This Calculator
Printable Formula Sheet
A one-page reference with every annulus formula on this page.
Annulus Formula Sheet
MegaCalcOnline.com · Ring area, width, radius and circumference formulas
Scan for the live calculator
| Find | Formula |
|---|---|
| Annulus area | A = π(R²−r²) |
| Ring width | w = R−r |
| Outer circumference | Co = 2πR |
| Inner circumference | Ci = 2πr |
| Mean radius | Rm = (R+r)/2 |
| Mean diameter | Dm = R+r |
| Outer radius from area+inner | R = √(r²+A/π) |
| Inner radius from area+outer | r = √(R²−A/π) |
| Pipe inner radius | r = OD/2 − t |
References
- Khan Academy — Circle geometry
- OpenStax Mathematics — area and measurement
- Wolfram MathWorld — Annulus
- NIST — SI units for length and area
- Maths Is Fun — Annulus Definition