How do you calculate the moment of inertia of a rectangle?

The moment of inertia of a rectangle about its centroidal horizontal axis is Ix = bh³/12, where b is the width and h is the height. About the vertical axis, Iy = hb³/12. For example, a 50 mm × 100 mm rectangle has Ix = 50 × 100³/12 = 4,166,667 mm⁴.

What is the parallel axis theorem?

The parallel axis theorem states that the moment of inertia about any axis parallel to the centroidal axis is I = Ic + Ad², where Ic is the centroidal moment of inertia, A is the cross-sectional area, and d is the perpendicular distance between the two axes. This theorem is essential for calculating moments of inertia of composite sections.

What is section modulus and why is it important?

Section modulus S = I/c, where I is the moment of inertia and c is the distance from the neutral axis to the extreme fibre. It is used directly in beam design with the flexure formula σ = M/S, where M is the bending moment. A larger section modulus means the beam can resist a greater bending moment before yielding.

Moment of Inertia — Simulation Trainer

Ix, Iy, Section Modulus, Radius of Gyration • 8 Cross-Sections • Parallel Axis Theorem — Simulate • Explore • Practice • Quiz

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Cross-Section Shape

Torque & Rotational Motion Simulator — Moment of Inertia Calculator

The moment of inertia (also called the second moment of area) is one of the most important geometric properties in structural and mechanical engineering. It quantifies how a cross-section's area is distributed relative to an axis, directly determining a beam's resistance to bending. A larger moment of inertia means greater stiffness and lower deflection under load. Engineers use moment of inertia calculations daily when designing beams, columns, shafts, and structural frames.

This free moment of inertia calculator lets you compute Ix, Iy, section modulus (Sx, Sy), radius of gyration (rx, ry), and polar moment of inertia (J) for eight standard cross-sections: rectangle, circle, hollow circle, hollow rectangle, I-beam, T-section, channel section, and angle section. The interactive canvas draws each shape to scale with dimension lines, centroid markers, and neutral axes, giving you an engineering-grade visualisation of the cross-section geometry.

How to Calculate Moment of Inertia for Common Shapes

For a rectangular cross-section with width b and height h, the moment of inertia about the centroidal horizontal axis is Ix = bh³/12. For a circular cross-section with diameter d, both Ix and Iy equal πd&sup4;/64. Hollow sections are calculated by subtracting the inner shape's moment of inertia from the outer. For example, a hollow circular tube has Ix = π(D&sup4; − d&sup4;)/64. The I-beam, widely used in structural steel construction, achieves a high Ix relative to its area by concentrating material in the flanges far from the neutral axis.

The Parallel Axis Theorem and Composite Sections

The parallel axis theorem states that I = Ic + Ad², where Ic is the centroidal moment of inertia, A is the area, and d is the distance between the centroid and the new axis. This theorem is essential when computing the moment of inertia of composite sections like T-beams and built-up sections, where individual component areas are offset from the overall centroid. The section modulus S = I/c connects moment of inertia to bending stress through the flexure formula σ = M/S, making it a direct measure of a beam's load-carrying capacity.

Section Modulus, Radius of Gyration, and Design Applications

The radius of gyration r = √(I/A) represents the distance from the axis at which the entire area could be concentrated to produce the same moment of inertia. It is critical in column buckling analysis using Euler's formula Pcr = π²EI/L², where a higher radius of gyration means greater resistance to buckling. In shaft design, the polar moment of inertia J determines torsional rigidity through the relationship τ = Tc/J, where T is the applied torque. Understanding these properties allows engineers to optimise cross-sections for weight, strength, and stiffness.

Who Uses This Simulator?

This moment of inertia calculator is designed for mechanical and civil engineering students, structural analysis trainees, strength of materials students, and instructors teaching cross-section properties, beam design, and column analysis. It provides instant visual feedback and precise numerical results without requiring textbook lookup tables or complex spreadsheets.

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If you found this Moment of Inertia simulator helpful, explore our Beam Bending Calculator, Shaft Torsion Simulator, Truss Analysis Simulator, and Pressure Vessel Calculator for more hands-on practice.