Calculate the area, width, inner/outer circumference, and average radius of an annulus (ring shape) from outer and inner radii. Includes presets for washers, rings, and pipes.
An annulus is the ring-shaped region between two concentric circles — one with a larger outer radius R and one with a smaller inner radius r. The area of this ring is π(R² − r²), which can also be written as π(R + r)(R − r). You encounter annular shapes constantly in everyday life and engineering: washers, pipe cross-sections, gaskets, CD/DVD surfaces, archery targets, circular running tracks, and the view through a telescope with a secondary mirror.
Calculating the annulus area is essential in materials engineering (how much material is in a pipe wall?), manufacturing (how much stock to cut for a washer?), and construction (how much paint for a circular border?). The width of the annulus (R − r) determines structural strength in pipes and tubes, while the average radius (R + r)/2 appears in the thin-shell approximation used in mechanical engineering.
There is an elegant alternative if you only know the width w of the annulus and the tangent length d (the length of a chord of the outer circle that is tangent to the inner circle), the area equals πd²/4. This surprising result does not require knowing either radius individually.
This calculator takes the outer and inner radii (with selectable length units), computes the annulus area, width, inner and outer circumferences, average radius, and the ratio of the annulus area to the outer circle area. Presets for common real-world rings (washers, pipes, O-rings) and a reference table make it practical for engineers, students, and DIY enthusiasts alike.
This annulus (ring) area calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Unit, Outer Radius (R), Inner Radius (r) and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Annulus area: A = π(R² − r²) = π(R + r)(R − r) Width: w = R − r Outer circumference: C_outer = 2πR Inner circumference: C_inner = 2πr Average radius: R_avg = (R + r) / 2 Average circumference: C_avg = 2π × R_avg Area ratio: A_annulus / A_outer = 1 − (r/R)² Tangent-length formula: A = π(d/2)² where d is the tangent chord
Result: For outer=6, inner=3.2, unit=mm, the tool returns the solved annulus (ring) area outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in annulus (ring) area formulas and reports derived values, checks, and classifications automatically.
This page is tailored to annulus (ring) area, with outputs tied directly to the form fields (Unit, Outer Radius (R), Inner Radius (r)). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
An annulus (plural: annuli) is the ring-shaped area between two concentric circles. It is also called an annular region or simply a ring.
Subtract the area of the inner circle from the outer circle: A = πR² − πr² = π(R² − r²). Use this as a practical reminder before finalizing the result.
R_avg = (R + r) / 2. It is the midpoint between the two radii and is used in the thin-shell approximation: A ≈ 2π R_avg × (R − r).
Yes — just divide diameters by 2 to get radii. Or equivalently, A = (π/4)(D² − d²) where D and d are the outer and inner diameters.
If a chord of the outer circle is tangent to the inner circle, its length d satisfies A = π(d/2)². You can calculate the annulus area without knowing R and r individually.
Washers, gaskets, pipe cross-sections, O-rings, CD/DVD writable surfaces, circular running tracks, and archery target rings are all annuli. Keep this note short and outcome-focused for reuse.