Annulus Area Calculator
Annulus (ring) area calculator with multiple input modes: inner/outer radius, diameters, or outer radius and thickness. Get ring area, circle areas, and an equivalent rectangle, with clear formulas and step-by-step explanation.
Full original guide (expanded)
Annulus Area Calculator
Compute the area of an annulus (circular ring) from inner/outer radii, diameters, or outer radius and thickness. The tool returns the ring area, individual circle areas, ring width, and an equivalent rectangle with the same area.
This calculator is designed for teaching, quick checks, and engineering back-of-the-envelope estimates. For safety-critical design, always validate with your organisation’s approved methods and software.
1. Select units and input mode
All dimensions must be expressed in the same unit. Areas are reported in the squared unit (e.g. cm²).
Input mode
2. Enter dimensions
Enter outer radius \( R \) and inner radius \( r \). The outer radius must be strictly larger than the inner radius.
3. Results
Outer radius \( R \)
Inner radius \( r \)
Ring thickness \( t = R - r \)
Outer circle area \( \pi R^2 \)
Inner circle area \( \pi r^2 \)
Annulus (ring) area
Equivalent rectangle
An annulus with area \( A = \pi(R^2 - r^2) \) has the same area as a rectangle with sides \( (R - r) \) and \( \pi(R + r) \). This visual identity is often used in geometry proofs.
Annulus (ring) area formula
An annulus is the region between two concentric circles with outer radius \( R \) and inner radius \( r \), with \( R > r \). Its area is simply the difference between the areas of the two disks:
If you prefer to work with diameters \( D = 2R \) and \( d = 2r \), you can rewrite the formula as:
Equivalent rectangle identity
Using the identity \( R^2 - r^2 = (R - r)(R + r) \), the annulus area can be written as:
Interpreting \( (R - r) \) as a width and \( \pi(R + r) \) as a height, we see that the annulus has the same area as a rectangle with those side lengths. Some geometric proofs literally cut an annulus into thin strips and rearrange them into such a rectangle.
Worked numerical example
Suppose an annular gasket has outer radius \( R = 10\ \text{cm} \) and inner radius \( r = 6\ \text{cm} \).
- Outer area: \( A_{\text{outer}} = \pi R^2 = \pi \cdot 10^2 = 100\pi \approx 314.16\ \text{cm}^2\).
- Inner area: \( A_{\text{inner}} = \pi r^2 = \pi \cdot 6^2 = 36\pi \approx 113.10\ \text{cm}^2\).
- Ring area: \( A = (100 - 36)\pi = 64\pi \approx 201.06\ \text{cm}^2\).
Using the rectangle identity:
- Ring thickness: \( t = R - r = 4\ \text{cm} \).
- \( R + r = 16\ \text{cm} \), so rectangle height \( = \pi(R + r) \approx \pi \cdot 16 \approx 50.27\ \text{cm} \).
- Rectangle area: \( 4 \cdot 50.27 \approx 201.06\ \text{cm}^2 \), matching the annulus area.
Engineering and practical considerations
- Consistent units: Always keep radii, diameters, and thickness in the same unit before applying any area formula. Mixing mm and cm without converting is a very common source of error.
- Tolerances: In mechanical design, manufacturing tolerances on inner and outer diameters can affect the effective area. For tight tolerance work, propagate tolerances through the formulas or use statistical tolerance methods.
- Cross-sectional area vs. projected area: In some applications (e.g. pipes, rings), the annulus area may refer to cross-sectional flow area; in others it may be a projected footprint. Make sure you are using the correct interpretation for your problem.
Frequently asked questions
Can the inner radius be zero?
Yes. If \( r = 0 \), the annulus reduces to a full disk of radius \( R \). The formula \( A = \pi(R^2 - r^2) \) still applies and becomes \( A = \pi R^2 \).
How do I compute ring thickness from area?
If you know \( A \), \( R \), and want to solve for \( r \), start from \( A = \pi(R^2 - r^2) \), so \( r^2 = R^2 - A/\pi \) and \( r = \sqrt{R^2 - A/\pi} \) (assuming \( R^2 \ge A/\pi \)). The current calculator focuses on forward problems (area from dimensions); for inverse problems you can rearrange the formulas as needed.
Why does the visual difference between the circles look small even if the area difference is large?
Human perception is better at judging linear dimensions than area. For example, doubling a radius quadruples the area. The ring formed by radii 9 and 10 may look thin, but its area is \( \pi(10^2 - 9^2) = 19\pi \), larger than a full disk of radius \( \sqrt{19} \).
How should I document an annulus area calculation in a report?
State the input dimensions (with units and tolerances), the formulas you used (for example \( A = \pi(R^2 - r^2) \)), and the numeric substitution. If the area feeds into further calculations (stress, flow, power), record intermediate results with sufficient significant figures so that colleagues can reproduce the results independently.
Formula (LaTeX) + variables + units
A_{\text{annulus}} = \pi R^2 - \pi r^2 = \pi\,(R^2 - r^2).
R = \frac{D}{2}, \quad r = \frac{d}{2}, \quad A_{\text{annulus}} = \frac{\pi}{4}\,(D^2 - d^2).
A_{\text{annulus}} = \pi(R^2 - r^2) = \pi(R - r)(R + r).
','\
\[ A_{\text{annulus}} = \pi R^2 - \pi r^2 = \pi\,(R^2 - r^2). \]
\[ R = \frac{D}{2}, \quad r = \frac{d}{2}, \quad A_{\text{annulus}} = \frac{\pi}{4}\,(D^2 - d^2). \]
\[ A_{\text{annulus}} = \pi(R^2 - r^2) = \pi(R - r)(R + r). \]
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.