Moment of Inertia Calculator
Calculate the area moment of inertia (second moment of area) for six common 2D shapes: rectangle, circle, triangle, ellipse, semicircle, and regular hexagon. Get Ix and Iy instantaneously.
About This Calculator
The Moment of Inertia Calculator (also known as the Second Moment of Area Calculator) is a structural engineering tool that computes the area moment of inertia for six common cross-sectional shapes. The area moment of inertia is a fundamental geometric property that measures how a cross-section's area is distributed about a given axis, directly determining a beam's resistance to bending and deflection.
For each shape, the calculator uses the standard centroidal formulas. For a rectangle with width b and height h, Ix = b·h^3/12 and Iy = h·b^3/12. For a solid circle of radius r, Ix = Iy = pi·r^4/4. For a triangle with base b and height h, Ix = b·h^3/36. The semicircle, ellipse, and regular hexagon each have their own specific formulas derived from integration of the area distribution.
The calculator supports six shapes: rectangle, circle, triangle (with adjustable apex offset for asymmetrical cases), semicircle, ellipse, and regular hexagon. Results display both Ix (about the x-axis) and Iy (about the y-axis) values with a visual bar chart comparison.
Regional Notes: This calculator is unit-agnostic and works globally. In India, dimensions are typically entered in mm or cm with results in mm^4 or cm^4. In the US, inches or feet are common with results in in^4. In the UK, mm or m are standard. No matter the unit system, the formulas remain identical since this is a pure geometric computation with no region-specific parameters.
Frequently Asked Questions
What is the area moment of inertia?
The area moment of inertia (also called the second moment of area) is a geometric property of a cross-section that measures how its area is distributed about an axis. It predicts a beam's resistance to bending and deflection. It has units of length^4 (e.g. mm^4, cm^4, in^4). Do not confuse it with the mass moment of inertia (kg·m^2) which relates to rotational dynamics.
What is the formula for moment of inertia of a rectangle?
For a rectangle with width b and height h, the area moment of inertia about the centroidal x-axis is Ix = b·h^3/12 and about the centroidal y-axis is Iy = h·b^3/12. For example, a 12 cm wide by 8 cm tall rectangle has Ix = 512 cm^4 and Iy = 1152 cm^4.
What is the difference between area moment of inertia and mass moment of inertia?
The area moment of inertia (second moment of area) describes how a cross-sectional area is distributed about an axis, used in beam bending and structural analysis. Its units are length^4 (e.g. m^4, in^4). The mass moment of inertia describes a body's resistance to rotational acceleration, used in rotational dynamics. Its units are mass·length^2 (e.g. kg·m^2, lb·ft·s^2).
How do I calculate the moment of inertia of a circle?
For a solid circle of radius r, the area moment of inertia about any centroidal axis is Ix = Iy = pi·r^4/4. For a circle of radius 5 units, the result is Ix = Iy ≈ 490.87 units^4.
Can this calculator handle different units?
Yes. Enter dimensions in any unit (mm, cm, m, in, ft) as long as all inputs use the same unit. The result will be in that unit raised to the fourth power. For example, if you enter cm, the result is in cm^4. The calculator is unit-agnostic and works consistently across all unit systems.
What shapes are supported by this calculator?
This calculator supports six common geometric shapes: rectangle, circle, triangle, semicircle, ellipse, and regular hexagon. Each shape uses the standard formula for area moment of inertia about its centroidal axes. For triangles, you can also set the apex offset for asymmetrical triangles.
How is the moment of inertia used in structural engineering?
In structural engineering, the area moment of inertia determines a beam's resistance to bending. A higher moment of inertia means less deflection under the same load. Engineers use it in beam deflection, stress analysis, and column buckling calculations. The I-beam shape is efficient because it concentrates area away from the neutral axis, maximizing the moment of inertia.
What is the parallel axis theorem?
The parallel axis theorem allows you to calculate the moment of inertia about any axis parallel to the centroidal axis. The formula is I = I_c + A·d^2, where I_c is the centroidal moment of inertia, A is the cross-sectional area, and d is the distance between the two parallel axes. This is essential for analyzing composite cross-sections.