Moment of Inertia (Rectangle) Calculator
The moment of inertia of a rectangular cross-section, more precisely its second moment of area, is the single most important geometric property for predicting how stiff a beam will be in bending. It describes how the cross-section's material is spread out relative to the axis it bends about, and it appears in every beam deflection and bending stress formula. For a solid rectangle bending about its horizontal centroidal axis, the value is the base times the height cubed, all divided by twelve. Because the height is cubed, making a beam deeper is far more effective than making it wider: doubling the height multiplies the moment of inertia by eight, while doubling the width only doubles it. This calculator takes the base and the height of a rectangular section and returns the second moment of area in the fourth power of your length unit, for example mm to the fourth when you enter millimeters. The result is shown to two decimal places. Section property conventions and related engineering standards are published by US federal agencies including the National Highway Traffic Safety Administration. Every figure is computed deterministically from the standard formula, shown below, with a worked example that reconciles exactly to the calculator so you can check each step yourself.
The second moment of area of a rectangle about its centroidal axis is b h^3 / 12. For a section 100 mm wide and 200 mm deep, the moment of inertia is 66,666,666.67 mm4. Depth matters most because height is cubed.
Rectangular moment of inertia formula
I = b h^3 / 12
I = second moment of area (length^4)
b = base (width) of the rectangle
h = height (depth) measured along the bending axis
Cube the height, multiply by the base, then divide by twelve. The axis runs horizontally through the centroid, so h is the dimension in the direction the beam bends. Because h is cubed, depth dominates the result, which is why structural beams are tall and narrow rather than short and wide.
Worked example
A rectangular section is 100 mm wide (base) and 200 mm deep (height), bending about its horizontal centroidal axis.
- Cube the height: 200^3 = 8,000,000
- Multiply by the base: 100 x 8,000,000 = 800,000,000
- Divide by 12: 800,000,000 / 12 = 66,666,666.67
- The moment of inertia is 66,666,666.67 mm4
So the rectangular section has a second moment of area of 66,666,666.67 mm4. These are the calculator's default inputs, so the result above matches the widget exactly.
Moment of Inertia (Rectangle) Calculator: frequently asked questions
How do you calculate the moment of inertia of a rectangle?
For a solid rectangle bending about its horizontal centroidal axis, the second moment of area is I = b h^3 / 12, where b is the base and h is the height. A 100 mm by 200 mm section gives 100 x 200^3 / 12 = 66,666,666.67 mm4.
Why is height cubed but base only linear?
Bending stiffness depends on how far material sits from the neutral axis, and that distance is set by the height. Material near the top and bottom contributes the most, so increasing depth has a cubed effect, while widening the section only adds material at a fixed distance and so scales linearly.
What is the difference between moment of inertia and second moment of area?
In structural mechanics the term moment of inertia usually means the second moment of area, a purely geometric property with units of length to the fourth power. It is distinct from the mass moment of inertia used in dynamics, which has units of mass times length squared.
Which axis does this formula use?
It uses the horizontal axis through the centroid, so h is the depth in the bending direction. To bend about the vertical axis instead, swap b and h, giving I = h b^3 / 12.
What is the rectangular moment of inertia formula?
The second moment of area of a rectangle about its centroidal axis is I = b h^3 / 12.
Official sources
- Structural section properties and engineering standards reference: US National Highway Traffic Safety Administration (NHTSA). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.