About

Aperture area determines the effective open cross-section of an opening: a lens diaphragm, a duct grille, a window frame, or a nozzle orifice. Getting it wrong propagates errors into flow-rate estimates, light-gathering calculations, and structural load analyses. This calculator computes area A for six standard shapes - circle, rectangle, ellipse, regular polygon, annulus, and circular sector - using exact closed-form equations. All dimensions are validated against physical constraints (positive, non-zero) before computation. Results are presented in five unit systems from mm² to ft². The tool assumes planar, Euclidean geometry and does not account for curvature or chamfer effects at edges.

Formulas

Each shape uses a distinct closed-form equation. The calculator selects the appropriate formula based on user-chosen geometry.

Circle

A = π ⋅ d²4

where d = diameter of the circular aperture.

Rectangle

A = w × h

where w = width, h = height.

Ellipse

A = π ⋅ a ⋅ b

where a = semi-major axis, b = semi-minor axis.

Regular Polygon

A = n ⋅ s²4 ⋅ tan(π ÷ n)

where n = number of sides (n ≥ 3), s = side length.

Annulus (Ring)

A = π ⋅ (R² − r²)

where R = outer radius, r = inner radius. Constraint: R > r.

Circular Sector

A = θ360 ⋅ π ⋅ r²

where θ = central angle in degrees (0 < θ ≤ 360), r = radius.

Unit conversion applies a multiplicative factor to convert the computed area in input units² into the target unit². For example, to convert from mm² to in², divide by 645.16.

Reference Data

Shape	Required Inputs	Formula	Typical Use Case
Circle	Diameter d	A = πr²	Lens aperture, pipe bore
Rectangle	Width w, Height h	A = w × h	Window opening, vent grille
Ellipse	Semi-major a, Semi-minor b	A = πab	Oval port, elliptical mirror
Regular Polygon	Sides n, Side length s	A = ns²4tan(π÷n)	Hexagonal cell, octagonal frame
Annulus (Ring)	Outer R, Inner r	A = π(R² − r²)	Washer, pipe cross-section
Circular Sector	Radius r, Angle θ	A = θ360 πr²	Fan blade, pie-cut opening
Unit Conversion Factors (relative to mm²)
mm²	1
cm²	1 cm² = 100 mm²
m²	1 m² = 1,000,000 mm²
in²	1 in² = 645.16 mm²
ft²	1 ft² = 92,903.04 mm²
Common Aperture Sizes
Camera f/2.8 (50 mm lens)	Effective diameter ≈ 17.86 mm		Area ≈ 250.5 mm²
Standard door (US)	914 × 2032 mm		Area ≈ 1.86 m²
6" pipe (SCH 40)	ID = 154.1 mm		Area ≈ 18,637 mm²
Telescope (8" Newtonian)	Diameter = 203.2 mm		Area ≈ 32,429 mm²
HVAC Register (10×6)	254 × 152.4 mm		Area ≈ 38,710 mm²

Frequently Asked Questions

As the number of sides n approaches infinity, the regular polygon converges to a circle. At n = 100, the area is within 0.05% of πr² where r is the apothem. The calculator supports n from 3 to 100.

The calculator validates that R > r before computing. If r ≥ R, the tool displays an error because the annular area would be zero or negative, which is physically meaningless.

Yes. For a camera lens, the effective aperture diameter equals the focal length divided by the f-number: d = f ÷ N. Enter that diameter in the circle mode. For a 50 mm lens at f/2.8, d ≈ 17.86 mm, yielding A ≈ 250.5 mm².

Conversion factors use the exact international definition: 1 in = 25.4 mm (exact). Therefore 1 in² = 645.16 mm² (exact). No rounding is introduced in the conversion step itself; rounding applies only to the displayed decimal places.

No. A circular sector is defined for 0 < θ ≤ 360°. Values outside this range are rejected. If you need to model overlapping sectors (e.g., a spiral), compute individual sectors and sum their areas manually.

Confusing diameter with radius is the single most common source of factor-of-four errors in aperture calculations. The circle input uses diameter because that is the dimension most commonly measured or specified in datasheets (pipe nominal diameter, lens aperture diameter). The annulus and sector inputs use radius because their formulas reference radii directly.