# Moment of Inertia Calculator Calculate the second moment of area (I) and section modulus (Z) for common cross-sections: solid circle, rectangle, hollow tube, and I-beam. Results in mm⁴ and cm⁴. > Canonical page: https://elysiatools.com/en/tools/moment-of-inertia-calculator - **Category:** Math & Numbers - **Keywords:** moment of inertia, second moment of area, section modulus, I-beam, beam, bending, mechanical engineering ## Overview The Moment of Inertia Calculator computes the second moment of area (I) and section modulus (Z) for common structural cross-sections, including solid circles, rectangles, hollow tubes, and I-beams. It provides instant, precise calculations in both mm⁴ and cm⁴ to assist mechanical and structural engineers in evaluating bending resistance and beam deflection. ## Inputs - **Cross-Section Shape** (select) - **Diameter D (mm)** (number): Diameter. Required for Solid Circle. - **Width b (mm)** (number): Section width (horizontal). Required for Rectangle. - **Height h (mm)** (number): Section height (vertical). Required for Rectangle. - **Outer Diameter Do (mm)** (number): Outer diameter. Required for Hollow Circle. - **Inner Diameter Di (mm)** (number): Inner diameter. Required for Hollow Circle. - **Flange Width B (mm)** (number): Flange width. Required for I-Beam. - **Total Height H (mm)** (number): Total height. Required for I-Beam. - **Web Thickness tw (mm)** (number): Web thickness. Required for I-Beam. - **Flange Thickness tf (mm)** (number): Flange thickness. Required for I-Beam. - **Decimal Places** (number) ## When to use - When designing structural beams and needing to calculate the second moment of area to predict deflection under load. - When sizing mechanical shafts, tubes, or pins and requiring the section modulus to determine bending stress limits. - When comparing different cross-sectional shapes to optimize material usage and weight. ## How it works - Select the cross-section shape from the options: Solid Circle, Rectangle, Hollow Circle (Tube), or I-Beam. - Input the required dimensions in millimeters, such as diameter, width, height, or flange and web thicknesses. - Set the desired number of decimal places for the output precision. - Generate the results to view the second moment of area (I), section modulus (Z), and total cross-sectional area. ## Use cases - Calculating the bending stiffness of a rectangular wooden beam for a floor joist design. - Determining the section modulus of a hollow steel tube to verify it can withstand a specific bending moment. - Evaluating the structural capacity of a custom-sized steel I-beam under vertical loading. ## Frequently asked questions ### What shapes does this calculator support? It supports solid circles, rectangles, hollow circles (tubes), and I-beams. ### What units are used for the inputs and outputs? Inputs are entered in millimeters (mm). Outputs are provided in square millimeters (mm²) for area, mm⁴ and cm⁴ for the second moment of area, and mm³ for the section modulus. ### How is the section modulus (Z) calculated? It is calculated as the second moment of area (I) divided by the maximum distance from the neutral axis to the outer fiber. ### Can I calculate the properties of a hollow rectangular section? No, this calculator currently supports solid rectangles, solid circles, hollow circular tubes, and standard I-beams. ### What is the difference between the second moment of area and the polar moment of inertia? The second moment of area measures resistance to bending, while the polar moment of inertia measures resistance to torsion. 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