Compute essential geometric properties of rectangular cross-sections: area, moments of inertia (Ix, Iy), elastic section moduli (Sx, Sy), radii of gyration, and polar moment.
The rectangle section is the most common cross-section in structural and mechanical engineering — from steel beams and timber joists to machine components. Its geometric properties directly govern bending stiffness, strength, and deflection behaviour. The area moment of inertia (second moment of area) Ix and Iy quantify resistance to bending about the horizontal (x) and vertical (y) centroidal axes, respectively. For a rectangle of base b and height h:
A = b · h
Ix = b · h3⁄12 , Iy = h · b3⁄12
Sx = Ix⁄(h/2) = b · h2⁄6 , Sy = h · b2⁄6
rx = √(Ix/A) = h/√12 , ry = b/√12
These formulas originate from classical beam theory (Euler–Bernoulli) and are derived by integrating the squared distance from the neutral axis. The elastic section modulus Sx and Sy directly relate to maximum bending stress: σmax = M / S. Engineers rely on these properties for safe dimensioning, code compliance (AISC, Eurocode), and optimisation of weight vs. strength.
The moment of inertia about the horizontal centroidal axis is obtained by integrating over the area: Ix = ∫ y² dA. For a rectangle, dA = b·dy, with y ranging from –h/2 to h/2, giving Ix = b·[y³/3]-h/2h/2 = b·h³/12. The parallel axis theorem allows transferring to any parallel axis. The section modulus represents the first moment of area relative to the extreme fibre. The radius of gyration is a fictitious radius that concentrates area to produce the same moment of inertia, crucial for buckling analysis (Euler column formula).
The polar moment of inertia J = Ix + Iy = b·h·(b²+h²)/12, which appears in torsion analysis of non-circular sections (St. Venant torsion constant for rectangles is approximated differently; however J is the polar second moment used in some contexts). For thin-walled open sections, the torsion constant differs, but for solid rectangles Jtorsion ≈ β·b·h³ (where β depends on aspect ratio), while our calculator reports Ix+Iy for completeness.
A structural engineer needs to select a rectangular hollow section (RHS) but for a solid rectangular timber beam of width 100 mm and height 250 mm. Bending occurs about the strong axis (height vertical). Ix = (100·250³)/12 = 130.2×10⁶ mm⁴. The maximum bending moment M = 15 kN·m, the bending stress σ = M·(h/2)/Ix = 15e6·125 / 130.2e6 ≈ 14.4 MPa, safe for timber grade C24. The tool instantly verifies Sx = 1.041×10⁶ mm³. Furthermore, for serviceability deflection control, the stiffness EIx governs. The rectangle calculator delivers all necessary parameters to iterate dimensions or compare alternatives.
The rectangle section also serves as the foundation for composite shapes (I-beams, T-sections) via the parallel axis theorem. Understanding the rectangular distribution is the first step toward advanced structural modeling. This calculator also supports design validation: check if your beam's section modulus meets required bending strength given yield stress.