円環（中空円）の面積計算機
同心円で囲まれた円環、フランジ、ワッシャーの正確な断面積を計算します。
Hollow Circle Area Calculator
Inner hole radius (must be less than outer radius).
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.
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.
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.
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!
Algebraic Proof: Area of an Annulus
Mathematical derivation of the hollow circle formulas:
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.
Heavy-Duty Steel Washer Clamping Face
A custom steel washer has OD = 60 mm and ID = 24 mm:
Hollow Concrete Bridge Pier
A seismic hollow round bridge column has OD = 3.0 m and wall thickness 0.4 m:
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:
Circular Running Track Surfacing Area
A circular warm-up running ring has inner boundary radius 40 m and track width 8 m:
Common Mistakes in Hollow Circle Calculations
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!
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.
Lay a straight rod tangent to the inner circle touching outer rim on both sides (length L). Area is immediately (π L²) / 4.
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.
円環に関するよくある質問
円の面積の求め方、計算公式、単位換算に関する主な質問と回答
円環の面積の公式は何ですか？
外半径 R と内半径 r から：面積 = π × (R² - r²)。外径 D と内径 d から：(π / 4) × (D² - d²)。
専門的な円の計算ツール
幾何学的パラメータに応じた最適な円の専門計算ツールを選択してください
Area of Circle Calculator
Find circle area from radius, diameter, or circumference with instant formulas.
直径から円の面積を計算するツール
Calculate circle area directly from diameter using A = (πd²)/4.
円周から円の面積を計算するツール
Determine circle area directly from perimeter using A = C²/(4π).
半円の面積計算機
Semicircle area, curved arc length, and total perimeter with flat base.
扇形と弓形の面積計算機
Circular sector and segment area from radius and central angle.
円の断面積計算機
Engineering cross-section for circular shafts, rebar, cables & tensile stress.
円形パイプの面積計算機
Internal fluid flow area vs solid pipe wall metal cross-sectional area.
積分による円の面積の求め方
Calculus derivation of circle area with polar, Cartesian, and shell integrals.