Annulus area calculator

Definition

A circular annulus is the set of plane points lying between two concentric circles of radii R and r, with R > r > 0 (the calculator also accepts the limiting cases R = r, area 0, and r = 0, a full disk). Its area is A = π(R² − r²) = π(R − r)(R + r). That is the difference of the areas of two disks: the larger of radius R and the smaller of radius r. This calculator accepts both radii (or the page’s equivalent inputs) and returns A.

The condition R > r guarantees positive area; when R = r the annulus degenerates to a circle of planar area zero. The centres coincide — an eccentric ring (distinct centres) needs integration or lens formulas and is not this calculator’s model.

Historical and conceptual context

Disk area A = πρ² is Archimedean (Measurement of a Circle): the disk equals a triangle with legs ρ and the circumference. Subtracting two such areas yields A_annulus = πR² − πr² at once. Euclid’s Elements prepare the result via similarity and exhaustion without modern algebra.

In applications the annulus models a pipe cross-section, a washer, a bearing race, or a zone between two isolines on a map. The formula computes planar area, not torus volume and not mass — a third-dimension thickness requires a separate multiplication by height.

Formal properties

Domain of a proper annulus: R > r > 0; the form accepts R ≥ r ≥ 0. The map A(R, r) increases in R and decreases in r. Homogeneity: under length scaling by λ > 0, area scales as λ². The identity A = π(R − r)(R + r) is convenient when the ring width w = R − r and the sum of radii are known.

Mean radius (R + r)/2 and width w give A = 2π·((R+r)/2)·w = π(R+r)w — exact algebraically for a concentric annulus, interpretable as a “rectangle” of length equal to the mid-circle circumference and width w.

As r → 0⁺ with fixed R, A → πR² (full disk). As R → r⁺, A → 0⁺.

Full derivation

Step 1. Area of a disk of radius R: A_R = πR². Area of a disk of radius r: A_r = πr².

Step 2. The annulus is the set difference of the disks (common centre), so A = A_R − A_r = π(R² − r²).

Step 3 (polar integral). ∫₀²π ∫_r^R ρ dρ dθ = 2π · (1/2)(R² − r²) = π(R² − r²).

Step 4 (factorisation). R² − r² = (R − r)(R + r), hence A = π(R − r)(R + r). This form helps when wall thickness w = R − r is the measured datum.

Special cases

For R = 2, r = 1: A = π(4 − 1) = 3π. For R = 10 cm, r = 8 cm (2 cm wall): A = π(100 − 64) = 36π cm² ≈ 113.1 cm². When r = R/2: A = (3/4)πR² — three quarters of the outer disk.

If diameters D and d are supplied instead of radii, A = (π/4)(D² − d²) for D > d > 0. Confusing diameter with radius scales each πρ² term by four.

Numerical and computational notes

Evaluating π·(R² − r²) is stable when R and r are not nearly equal at floating-point range. When R ≈ r at large magnitude, the factorisation (R−r)·(R+r) reduces cancellation loss from subtracting close squares.

Input with r ≥ R or non-positive values is rejected. Comma and period are equivalent. The calculator does not model an elliptical ring or an annular sector.

In mechanics of materials an annular cross-section appears for pipes, hollow shafts, and sealing rings. The second moment of area of an annulus about its centre is (π/4)(R⁴ − r⁴) — that is not the area, but a quantity obtained by integrating ρ³. This calculator returns only A = π(R² − r²).

Ring width w = R − r and mean circumference 2π·((R+r)/2) multiply to the area: A = w · π(R+r). That identity helps when a micrometer measures wall thickness and a tape measures a mean or outer girth. One must still convert consistently to a pair (R, r) before substituting.

As a figure with fixed radii: once R and r are fixed, the concentric annulus has no further shape freedom. For fixed A many pairs (R, r) satisfy R² − r² = A/π — the calculator does not choose a pair; it only computes A from the radii supplied.

Historically the Latin annulus (“small ring”) is standard in English texts; school Polish usually says “pierścień kołowy”, with “wycinek pierścienia” for an annular sector. MathWorld and Wikipedia (Annulus) give the same π(R² − r²) formula and the difference-of-disks reading.

Polar coordinates make the derivation transparent: the area element ρ dρ dθ integrates in ρ from r to R and in θ over a full turn. Restricting θ to an interval of length α yields an annular sector of area (α/(2π))·π(R² − r²) = (α/2)(R² − r²). That sector model is separate from the full-annulus calculator.

Related concepts and authorities

Related models: circle area, circumference, sector area, and pipe volume (annulus × height). Annulus area is not the surface area or volume of a torus.

Authorities: Archimedes, Measurement of a Circle; Euclid, Elements; MathWorld — Annulus; Wikipedia — Annulus (mathematics); NIST DLMF (π). The calculator is educational.