Section properties reference

Every section property the Section Analyzer reports — area, centroid, second moments of area, elastic and plastic section moduli, radii of gyration, principal axes, polar moment, shape factors, and stiffness-weighted (transformed) properties — with formulas, units, and where they come from.

How to read this page

Every property reported in the right sidebar (and in the JSON response from the analyser) has an entry here. Each entry gives:

The standard engineering symbol
The formula the analyser uses
The units in SI/metric (always)
The JSON path in the response (so you can wire it up to a report or an external integration)
A short note on what the property is for

The shape of the response — the exact field names and nesting — is defined by the SectionProperties type:

All properties are reported about the centroidal axes of the section unless explicitly stated otherwise.

Sign and axis conventions

GCS follows the platform-wide canonical:

Engineer-Y-up: the cross-section is drawn with $y$ positive upward. yMax is the physical top of the section; yMin is the physical bottom. xMin is the left face; xMax is the right face.
$I_{xx} = \int y^2\,dA$ : second moment of area about the $x$ -axis (its kernel is $y^2$ , so it scales with how tall the section is). $I_{yy}$ is the analogous integral about the $y$ -axis.
$S_{x,\mathrm{top}} = I_{xx}/(y_\mathrm{max} - \bar{y})$ : the elastic section modulus at the physical top fibre of the section as drawn. Sx_bot, Sy_left, Sy_right follow the same physical-face naming. The names are truthful — Sx_top is never aliased to a coordinate extreme.
Plastic moduli $Z_x$ , $Z_y$ are about the centroidal axes (the lines that split the section into equal areas).

For the full canonical (moment signs, stress signs, biaxial loading) see Platform — Sign and axis conventions. GCS itself does not consume applied loads — it reports section properties only — but downstream tools (ACS, LGS) that consume these section moduli for stress calculations follow the canonical strictly.

Geometric properties

Area

Item	Value
Symbol	$A$
Units	mm²
Formula	$A = \displaystyle\oint_{\partial\Omega} x\,\mathrm{d}y$ (shoelace for straight edges) + analytic circular-segment corrections for arc edges
JSON path	`$.area`

The cross-sectional area, computed by the shoelace formula over the polygon boundary. Voids subtract; inclusions do not change the gross area (they replace the host material’s contribution at the same footprint).

Perimeter

Item	Value
Symbol	$P$
Units	mm
Formula	$P = \displaystyle\sum_i \lvert \mathbf{p}_{i+1} - \mathbf{p}_i \rvert$
JSON path	`$.perimeter`

Total length of all edges across all regions.

Centroid

Item	Value
Symbol	$(\bar{x}, \bar{y})$
Units	mm
Formula	$\bar{x} = \dfrac{1}{6A}\oint x^2\,\mathrm{d}y$ , $\bar{y} = -\dfrac{1}{6A}\oint y^2\,\mathrm{d}x$
JSON path	`$.centroid.x`, `$.centroid.y`

The geometric centre of the section. The neutral axis under pure bending of a homogeneous elastic section passes through the centroid. For composite sections, the geometric centroid is reported here; the transformed centroid (the elastic neutral-axis crossing for the composite) is reported under $.transformed.centroid.

Second moments of area (centroidal)

Item	Value
Symbols	$I_{xx}$ , $I_{yy}$ , $I_{xy}$
Units	mm⁴
Formula	Green’s theorem on the chord polygon + closed-form circular-segment correction terms for arc edges
JSON path	`$.momentOfInertia.Ix`, `$.momentOfInertia.Iy`, `$.momentOfInertia.Ixy`

The second moments of area about the centroidal $x$ and $y$ axes, and the product of inertia $I_{xy}$ . For sections with two axes of symmetry $I_{xy} = 0$ exactly; for unsymmetric sections it is nonzero and drives the principal-axis rotation $\theta$ below.

When the section contains arc edges the calculation uses exactRingProperties — a closed-form algorithm that delivers exact results (zero discretisation error) regardless of arc sweep angle. The winding direction is detected from the cross-sum so $I_{xy}$ carries the correct sign without forcing callers to wind their polygons CCW.

Extreme fibres

Item	Value
Symbols	$y_\mathrm{top}$ , $y_\mathrm{bot}$ , $x_\mathrm{left}$ , $x_\mathrm{right}$
Units	mm (distance from centroid)
JSON path	`$.extremeFibers.yTop`, `.yBot`, `.xLeft`, `.xRight`

The signed perpendicular distances from the centroid to the section’s extreme fibres along the $x$ and $y$ axes. Used to compute elastic section moduli at each fibre.

Elastic section moduli

Item	Value
Symbols	$S_{x,\mathrm{top}}$ , $S_{x,\mathrm{bot}}$ , $S_{y,\mathrm{left}}$ , $S_{y,\mathrm{right}}$
Units	mm³
Formula	$S = I / c$ where $c$ is the extreme-fibre distance
JSON path	`$.elasticModulus.Sx_top`, `.Sx_bot`, `.Sy_left`, `.Sy_right`

The elastic section modulus at each of the four extreme fibres. Used to convert an applied bending moment to extreme-fibre stress: $\sigma = M/S$ .

Four values are reported (rather than two) so unsymmetric sections — where the centroid is not equidistant from top/bottom or left/right — carry the right $S$ for each face. For a doubly symmetric section all four collapse to two pairs.

Plastic section moduli

Item	Value
Symbols	$Z_x$ , $Z_y$
Units	mm³
Formula	$Z = \displaystyle\int_{A_\mathrm{half}} y\,\mathrm{d}A$ about the plastic neutral axis (each axis solved by polygon clipping + binary search)
JSON path	`$.plasticModulus.Zx`, `.plasticModulus.Zy`

The plastic section modulus about each centroidal axis, computed by polygon clipping. The plastic neutral axis is the line that splits the section into two equal-area halves; $Z$ is the sum of the first moments of those halves about that axis.

Null for composite sections. When the section contains more than one material the plastic modulus is null and plasticModulusReason carries an explanation. See Materials — Plastic modulus is null for composite sections.

Radii of gyration

Item	Value
Symbols	$r_x$ , $r_y$
Units	mm
Formula	$r = \sqrt{I / A}$
JSON path	`$.radiusOfGyration.rx`, `.ry`

The radius of gyration about each centroidal axis — a length-scale that combines stiffness and area. Used in classical column slenderness ratios $L/r$ .

Principal moments

Item	Value
Symbols	$I_1$ , $I_2$ , $\theta$
Units	mm⁴, mm⁴, degrees
Formula	Mohr’s-circle decomposition: $I_{1,2} = (I_{xx}+I_{yy})/2 \pm \sqrt{((I_{xx}-I_{yy})/2)^2 + I_{xy}^2}$ ; $\theta = \tfrac{1}{2}\tan^{-1}\big(2 I_{xy}/(I_{xx}-I_{yy})\big)$
JSON path	`$.principalMoments.I1`, `.I2`, `.theta`

The principal moments are the eigenvalues of the inertia tensor; $\theta$ is the angle from the geometric $x$ -axis to the major principal axis (counterclockwise positive). For doubly symmetric sections $\theta$ is 0 (or 90 — they are equivalent) and $I_1 = I_{xx}$ , $I_2 = I_{yy}$ .

Polar moment

Item	Value
Symbol	$J_o$
Units	mm⁴
Formula	$J_o = I_{xx} + I_{yy}$ (about the centroid)
JSON path	`$.polarMoment`

The polar second moment about the centroidal axis. This is the classical polar moment ( $I_{xx} + I_{yy}$ ), not the St. Venant torsion constant $J$ . For open thin-walled sections the two differ substantially — if you need $J$ for torsion design use LGS, which computes it from the mid-line via Vlasov beam theory.

Shape factors

Item	Value
Symbols	$k_x$ , $k_y$
Units	dimensionless
Formula	$k = Z / S$
JSON path	`$.shapeFactor.kx`, `.ky`

The ratio of plastic to elastic section modulus about each axis. Common textbook values: 1.5 for rectangles, 1.7 for solid circles, ~1.12 for I-sections about their strong axis. The shape factor characterises how much reserve capacity the section has beyond first-fibre yield — a measure of post-elastic stiffness.

Null for composite sections (same reason as plastic modulus).

Transformed (stiffness-weighted) properties

Reported only for composite (multi-material) sections via the modular-ratio method. Available under the $.transformed branch of the response.

Field	JSON path	Notes
Reference material	`$.transformed.referenceMaterial`	`{ id, name, elasticModulus_MPa }` of the material the transformed properties are referenced to (the first outline’s material)
Area	`$.transformed.area`	Stiffness-weighted equivalent area
Centroid	`$.transformed.centroid.x/y`	The elastic neutral-axis crossing of the composite (in general different from the geometric centroid)
Second moments	`$.transformed.momentOfInertia.Ix/Iy/Ixy`	Transformed inertias about the transformed centroid
Extreme fibres	`$.transformed.extremeFibers.yTop/yBot/xLeft/xRight`	About the transformed centroid
Elastic moduli	`$.transformed.elasticModulus.Sx_top/Sx_bot/Sy_left/Sy_right`	$S = I / c$ on the transformed section
Radii of gyration	`$.transformed.radiusOfGyration.rx/ry`	—
Principal moments	`$.transformed.principalMoments.I1/I2/theta`	—
Polar moment	`$.transformed.polarMoment`	—

The transformed property set deliberately omits plastic modulus and shape factor — see Materials — Plastic modulus is null for composite sections.

Where the numbers come from

The Section Analyzer has two computation paths that compute the same numbers from the same inputs:

Client (browser) — calculatePolygonProperties and calculateCompositeProperties in client/src/lib/math-utils.ts. Used for instant feedback while drawing. The sidebar badge says Preview.
Server (authoritative) — SectionPropertyCalculator.CalculateAll in server-dotnet/Core/GeneralSection/. Runs automatically as soon as the geometry validates server-side; there is nothing to click. The sidebar badge flips to Calculated when the server response arrives. The PDF report uses this same path — there is no third computation.

The two paths use the same algorithm for each property:

Arc-aware exact properties (area, centroid, $I_{xx}$ , $I_{yy}$ , $I_{xy}$ , extreme fibres, perimeter): When the section contains arc edges, exactRingProperties computes these quantities by combining Green’s theorem integrals on the chord polygon with analytic circular-segment correction terms for each arc. The result is exact to floating-point precision — no discretisation error regardless of arc sweep angle or radius.
Plastic moduli ( $Z_x$ , $Z_y$ ): Always computed by Sutherland-Hodgman polygon clipping + binary search on the plastic neutral axis. Arc edges are discretised to a fine polygon before clipping, so the plastic modulus carries a small (sub-0.1 %) discretisation error for curved sections.

Both paths are unit-tested against textbook results for rectangles, circles, I-sections, and a handful of composite scenarios.

Reference material

Closed-form polygon section-property derivations are textbook material. For background on the integrals and the validation cases:

Pilkey — Analysis and Design of Elastic Beams (the canonical reference for cross-section properties of arbitrary shapes)
Roark’s Formulas for Stress and Strain — tabulated section properties for a wide range of standard and unusual shapes; good for sanity-checking against
Park & Paulay — Reinforced Concrete Structures, Chapter 4 (transformed-section analysis for composite RC)

Next steps

Worked example: composite section — see every property of this page populated for a real composite section
Limitations — what the analyser does not compute (warping constant, St. Venant torsion, etc.) and where to go for them