Moment of Inertia Calculator

The moment of inertia (also called the second moment of area) is a geometric property that measures a shape's resistance to bending and deflection. Higher values mean the shape is stiffer and harder to bend.

Area moment of inertia (I) measures resistance to bending and is used in structural engineering. Mass moment of inertia (J) measures resistance to angular acceleration and is used in dynamics and rotational mechanics.

For a rectangle about its centroidal axis: I = bh³/12, where b is the base width and h is the height. About the base: I = bh³/3. The parallel axis theorem adjusts for off-center axes.

For a solid circle about its center: I = πr⁴/4. For a hollow circle (ring): I = π(R⁴ - r⁴)/4, where R is the outer radius and r is the inner radius.

The parallel axis theorem states: I = I_centroid + Ad², where I_centroid is the moment about the centroid, A is the area, and d is the distance between axes. This lets you shift from centroidal to any parallel axis.

The polar moment of inertia (J or I_p) measures resistance to torsion (twisting). For a circle: J = πr⁴/2. It equals I_x + I_y by the perpendicular axis theorem.

The radius of gyration k = √(I/A) represents the distance from the axis at which all the area could be concentrated to give the same moment of inertia. It is used in column buckling calculations.

Moment of inertia determines beam deflection (δ = PL³/48EI), bending stress (σ = My/I), and column buckling load. Higher I values mean less deflection and greater load capacity.

An I-beam's moment of inertia is calculated by treating it as three rectangles: I = (bh³ - (b-t_w)(h-2t_f)³)/12, where b is flange width, h is total height, t_w is web thickness, and t_f is flange thickness.

Area moment of inertia has units of length⁴ (e.g., m⁴, mm⁴, in⁴). Mass moment of inertia has units of mass × length² (e.g., kg·m², lb·ft²).

The section modulus S = I/c is the ratio of moment of inertia to the distance from the neutral axis to the extreme fiber. It is used to calculate maximum bending stress: σ_max = M/S.

Material farther from the axis increases I significantly (since I depends on distance squared). This is why I-beams and hollow tubes are efficient: they concentrate material at the top and bottom where bending stresses are highest.

Yes, the moment of inertia of a composite shape is the sum of its parts: I_total = I₁ + I₂ + I₃ + ... Use the parallel axis theorem to shift each component to the composite centroidal axis.

For a triangle about its base: I = bh³/12. About its centroid: I = bh³/36. About the apex: I = bh³/4. The height h is measured perpendicular to the base b.

Beam deflection is inversely proportional to I: larger I means less deflection. For a simply supported beam with a point load: δ = PL³/(48EI), where E is the elastic modulus and I is the moment of inertia.

The second moment of area is another name for the area moment of inertia. It is calculated as I = ∫y² dA, summing the product of each area element and its distance squared from the reference axis.

Use the centroidal axis (through the geometric center) for standard formulas. Use the parallel axis theorem to shift to other axes. For bending analysis, use the axis about which the bending moment acts.

For a hollow rectangle: I = (BH³ - bh³)/12, where B and H are outer dimensions and b and h are inner dimensions. This is common in box-section structural members.

In rotational dynamics, I determines angular acceleration: τ = Iα. A higher I means more torque is needed for the same angular acceleration. This is why figure skaters pull their arms in to spin faster.

For a semicircle about its diameter: I = πr⁴/8. About its centroid (which is offset by 4r/(3π) from the diameter): I = r⁴(π/8 - 8/(9π)).

The Moment of Inertia Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Moment of Inertia Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.