Pierścień Kołowy

Kalkulator Pierścienia Kołowego

Oblicz dokładne pole powierzchni pierścienia kołowego, kołnierza lub podkładki ze średnic zewnętrznej i wewnętrznej.

Annulus / Ring Geometry

Hollow Circle Area Calculator

Input Mode

Outer Radius (R)

Inner Radius (r)

Inner hole radius (must be less than outer radius).

A = π × (R² - r²) = (π/4) × (D² - d²)

Wall Thickness t = R - r

Hollow Circle / Annulus Area

0 cm²

Wall Thickness (t) 0

Outer Area 0

Hole Area 0

Concentric Annulus Rings Ring Area

Calculation Breakdown:

Enter outer and inner dimensions to calculate annulus area.

What Is a Hollow Circle (Annulus)? Anatomy & Core Formulas

An annulus is the planar region bounded by two concentric circles of different radii sharing a common center. From mechanical washers and bearings to pipe cross-sections and planetary rings, it is one of engineering's most fundamental shapes.

The material area is the difference between the larger outer circle and the smaller inner void. Because it has two circular edges, an annulus features both an outer perimeter (2πR) and an inner perimeter (2πr), giving it a total wetted perimeter of 2π(R + r).

Whether sizing bolt washers to prevent crushing wood or determining gasket contact stress on high-pressure steam pipes, calculating annulus area is the primary design step.

Annulus Geometric Parameters & Formulas

Parameter	Radii Formula (R, r)	Diameters Formula (D, d)	Example (R=10, r=6 cm)
Enclosed Ring Area (A)	A = π × (R² - r²)	A = (π / 4) × (D² - d²)	201.06 cm²
Wall Thickness (t)	t = R - r	t = (D - d) / 2	4.00 cm
Mean Radius (r_avg)	r_avg = (R + r) / 2	r_avg = (D + d) / 4	8.00 cm
Total Wetted Perimeter	P = 2π(R + r)	P = π(D + d)	100.53 cm
Tangent Chord Length (L)	L = 2√(R² - r²)	L = √(D² - d²)	16.00 cm
Polar Moment of Inertia (J)	J = (π / 2)(R⁴ - r⁴)	J = (π / 32)(D⁴ - d⁴)	13,671.86 cm⁴

4 Ways to Calculate Hollow Circle Area

Choose the mathematical approach best suited to your available tooling, calipers, or theoretical specs.

Radii Difference: A = π × (R² - r²)

The standard analytical method. Square the outer radius, square the inner radius, subtract them, and multiply by π.

For R = 10, r = 6: π × (100 - 36) = 64π ≈ 201.06 sq units

Diameters Difference: A = (π / 4) × (D² - d²)

Standard in machining and piping. Measure OD and ID with calipers: square OD, square ID, subtract, and multiply by 0.7854.

For D = 20, d = 12: 0.7854 × (400 - 144) ≈ 201.06

Centerline Wall Method: A = 2π × r_avg × t

Multiply the mean centerline perimeter by the radial wall thickness. Excellent for sheet metal rolls and thin-walled pipes.

For r_avg = 8 cm, t = 4 cm: 2π × 8 × 4 = 64π ≈ 201.06 cm²

Mamikon's Tangent Chord: A = (π × L²) / 4

Lay a straight ruler tangent to the inner circle bounded by the outer circle. The ring's area is exactly equal to a circle of diameter L!

For tangent chord L = 16 cm: (π × 16²) / 4 = 64π ≈ 201.06 cm²

Algebraic Proof: Area of an Annulus

Mathematical derivation of the hollow circle formulas:

1. Area of outer circle of radius R: A_outer = πR²

2. Area of inner circle hole of radius r: A_inner = πr²

3. Annulus Area = A_outer - A_inner = πR² - πr²

4. Factor out π: Area = π(R² - r²)

5. Substitute R = D/2 and r = d/2: Area = π[(D/2)² - (d/2)²] = π[D²/4 - d²/4]

6. Final Formula: Area = (π / 4) × (D² - d²) ≈ 0.785398 × (D² - d²) [Q.E.D.]

Standard Mechanical Washers & Rings Reference Table

Standard USS/SAE bolt washer sizes, outer/inner diameters, and contact areas.

Bolt / Washer Size	Outer Dia (D)	Inner Hole (d)	Wall Width (t)	Contact Area
1/4" SAE Washer	0.625 in	0.281 in	0.172 in	0.2447 sq in
3/8" SAE Washer	0.812 in	0.406 in	0.203 in	0.3884 sq in
1/2" SAE Washer	1.062 in	0.531 in	0.266 in	0.6640 sq in
3/4" SAE Washer	1.469 in	0.812 in	0.328 in	1.1769 sq in
M10 Metric Washer	20.0 mm	10.5 mm	4.75 mm	227.57 mm²
M16 Metric Washer	30.0 mm	17.0 mm	6.50 mm	479.88 mm²

4 Detailed Solved Real-World Examples

Realistic engineering, structural, gasket sealing, and sports facility calculations with exact steps.

Example 1: Mechanical Fasteners

Heavy-Duty Steel Washer Clamping Face

A custom steel washer has OD = 60 mm and ID = 24 mm:

D = 60 mm, d = 24 mm (R = 30, r = 12 mm)

D² - d² = 60² - 24² = 3,600 - 576 = 3,024 mm²

Area = (π / 4) × 3,024 = 756π ≈ 2,375.04 mm² (23.75 cm²)

Example 2: Civil Engineering

Hollow Concrete Bridge Pier

A seismic hollow round bridge column has OD = 3.0 m and wall thickness 0.4 m:

Outer R = 1.5 m, Inner r = 1.5 - 0.4 = 1.1 m

R² - r² = 2.25 - 1.21 = 1.04 m²

Concrete Cross-Section = π × 1.04 ≈ 3.267 m²

Saves 3.80 m² (53.8%) of concrete compared to a solid 3m column!

Example 3: Pipeline Flange Gasket

Steam Flange Gasket Sealing Stress

An ASME B16.21 gasket has OD = 10 inches and ID = 6 inches. Total bolt clamp load = 80,000 lbs:

D = 10 in, d = 6 in

Gasket Area = (π / 4) × (100 - 36) = 16π ≈ 50.27 sq in

Sealing Stress σ = 80,000 / 50.265 ≈ 1,591.55 psi

Example 4: Athletics Facility

Circular Running Track Surfacing Area

A circular warm-up running ring has inner boundary radius 40 m and track width 8 m:

r = 40 m, R = 48 m

R² - r² = 2,304 - 1,600 = 704 m²

Track Area = π × 704 ≈ 2,211.68 m²

Total Boundary Curbing = 2π(48 + 40) ≈ 552.92 linear meters

Common Mistakes in Hollow Circle Calculations

Subtracting Radii Before Squaring: π(R - r)²

Wrong: π × (R - r)². Correct: π × (R² - r²). For R=10 and r=6: π(10-6)² = 16π (50.3), but true area is π(100-36) = 64π (201.1). Subtracting first loses 75% of area!

Forgetting to Divide by 4 When Using Diameters

When using OD (D) and ID (d), you must divide by 4: (π / 4)(D² - d²). Omitting 4 gives a result 4 times too large.

Pro Tip: Mamikon's Visual Shortcut

Lay a straight rod tangent to the inner circle touching outer rim on both sides (length L). Area is immediately (π L²) / 4.

Pro Tip: Torsional Efficiency

A hollow drive shaft with ID = 0.75 OD has 90% of the torsional strength of a solid shaft while saving 56% of total weight.

Industry Applications of Hollow Circle Geometry

Mechanical Fasteners

Sizing Belleville disc springs, flat washers, and lock rings to distribute bolt clamping torque.

Flange & Gasket Sealing

ASME B16.5 pipe flange gasket contact face calculations and bolting pre-load verification.

Civil Bridge Columns

Designing hollow cylindrical concrete overpass pylons for seismic ductility and material reduction.

Automotive Brakes

Brake rotor swept annular friction area and thermal heat dissipation capacity during deceleration.

Optical Instrumentation

Camera lens aperture stops, telescope secondary mirror obstruction ring areas, and annular light shields.

Underground Utilities

Precast concrete manhole riser rings, dry well infiltration sidewalls, and deep foundation caisson rings.

Często zadawane pytania: Pierścień kołowy

Odpowiedzi na najczęstsze pytania dotyczące wzorów, jednostek i obliczeń koła

Jaki jest wzór na pole pierścienia kołowego?

Z promieniami: P = π × (R² - r²). Ze średnicami: P = (π / 4) × (D² - d²).

Wybierz narzędzie obliczeniowe dopasowane do Twoich parametrów geometrycznych

Main Tool

Area of Circle Calculator

Find circle area from radius, diameter, or circumference with instant formulas.

Kalkulator Średnicy

Kalkulator Pola Koła ze Średnicy

Calculate circle area directly from diameter using A = (πd²)/4.

Kalkulator Obwodu

Kalkulator Pola Koła z Obwodu

Determine circle area directly from perimeter using A = C²/(4π).

Geometria Półkola

Kalkulator Pola Półkola

Semicircle area, curved arc length, and total perimeter with flat base.

Wycinek i Odcinek Koła

Kalkulator Wycinka i Odcinka Koła

Circular sector and segment area from radius and central angle.

Inżynieria Konstrukcyjna

Kalkulator Pola Przekroju Poprzecznego Koła

Engineering cross-section for circular shafts, rebar, cables & tensile stress.

Hydraulika i Rurociągi

Kalkulator Przekroju Rury Okrągłej

Internal fluid flow area vs solid pipe wall metal cross-sectional area.

Dowód Całkowy

Pole Koła za Pomocą Całki

Calculus derivation of circle area with polar, Cartesian, and shell integrals.