Annulus Calculator

A washer, a pipe cross-section, a circular fountain basin, a ring beam foundation — all of these are annuli: the flat region trapped between two concentric circles. The calculator for annulus geometry finds area, perimeter, and width from any two of the three defining measurements (outer radius, inner radius, or width), handling the geometry that the basic circle calculator cannot.

Annulus Formulas: Area and Perimeter

For an annulus with outer radius R and inner radius r (where R > r ≥ 0):

Area = π(R² − r²) = π(R + r)(R − r)

Perimeter = 2π(R + r) — the sum of the outer and inner circumferences

Width (ring thickness) = R − r

The area formula is simply the outer circle area minus the inner circle area. Note that the annulus perimeter includes both the outer circumference (2πR) and the inner circumference (2πr) — the full boundary of the shape consists of two separate circles. For a washer with outer diameter 30 mm and inner diameter 20 mm: R = 15 mm, r = 10 mm; Area = π(225 − 100) = 392.7 mm²; Perimeter = 2π(15 + 10) = 157.1 mm. Use this online calculator for any annulus dimensions. The ellipse calculator covers the analogous elliptical ring geometry.

Engineering Applications: Pipe Cross-Sections and Hollow Cylinders

The annulus is the fundamental shape in piping and hollow structural analysis:

• Pipe flow area: the cross-sectional area available for fluid flow in a pipe equals the annulus area between the outer and inner walls — critical for velocity and flow rate calculations in hydraulics
• Hollow shaft strength: the second moment of area (moment of inertia) of a hollow circular section I = π(R⁴ − r⁴)/4; used in beam bending and torsion analysis
• Insulation volume: pipe insulation is a hollow cylinder; its cross-section is an annulus; material quantity per meter = π(R² − r²) × 1 m
• Washer and gasket sizing: mechanical engineers use the annulus area to calculate contact stress and sealing effectiveness

The regular polygon calculator and plane geometry calculators cover related geometric shapes for engineering and mathematical applications.

The Annulus in Architecture and Construction

Ring-shaped structures appear throughout architecture: circular foundations, donut-shaped reflecting pools, ring roads, and stadium track cross-sections all involve annulus geometry. A circular swimming pool with inner radius 5 m (pool area) and outer radius 6 m (including deck) has a deck annulus area = π(36 − 25) = 34.56 m² — useful for material quantity estimation and cost calculation. Ring beam foundations for circular tanks and silos distribute load around the perimeter; the annulus cross-sectional area determines the concrete volume per linear meter of beam.

Annulus Area as a Limit: From Ring to Disk

Two special cases of the annulus reveal its relationship to other shapes. When inner radius r → 0, the annulus becomes a full disk with area πR². When inner radius r → R (ring width → 0), the annulus becomes a thin circle (circumference) with area → 0. The annulus is therefore a generalization of both the disk and the circle, and its area formula π(R² − r²) interpolates continuously between these extremes. This makes the annulus a pedagogically useful shape for demonstrating the difference between area and perimeter as limiting cases of a parameterized family.