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Rectangle Moment of Inertia Calculator

Enter the width and height to find the rectangle's moment of inertia.

How it is calculated

Width (base) and height are entered in millimetres, and the results are in mm⁴, mm³ and mm². The tool gives the second moment of area about both centroidal axes of the rectangle: Ix = b·h³ ÷ 12 about the horizontal axis and Iy = h·b³ ÷ 12 about the vertical axis. The section moduli follow by dividing by the distance from the axis to the outer edge: Sx = Ix ÷ (h/2) and Sy = Iy ÷ (b/2). The area b·h is also shown. Both dimensions must be above zero. The axes pass through the centre of the rectangle, so the values do not apply to an axis along an edge. Enter other length units only if you accept results in that unit to the fourth or third power.

Worked example

A section 50 mm wide and 100 mm high. Ix = 50 × 100³ ÷ 12 = 50 × 1,000,000 ÷ 12 = 4,166,666.67 mm⁴. Iy = 100 × 50³ ÷ 12 = 100 × 125,000 ÷ 12 = 1,041,666.67 mm⁴. Section moduli: Sx = 4,166,666.67 ÷ 50 = 83,333.33 mm³ and Sy = 1,041,666.67 ÷ 25 = 41,666.67 mm³. The area is 5,000.00 mm². Standing the section tall gives four times the stiffness of laying it flat.

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Common mistakes

  • Swapping base and height. Bending about the horizontal axis uses the height cubed, so a 50 × 100 beam on edge and the same beam flat give very different Ix.
  • Entering centimetres or metres and reading the result as mm⁴. The unit is the one you typed, raised to the fourth power.
  • Using Ix for bending about the vertical axis. For a load that bends the section sideways, take Iy, which uses the base cubed.
  • Confusing the second moment of area with the mass moment of inertia. This one depends only on the cross-section shape, not on the material or the weight.
  • Using the formula for a hollow or I-section. This tool covers only a solid rectangle; a hollow box needs the inner rectangle's value subtracted.

About this calculator

The second moment of area (commonly called the moment of inertia of a cross-section, though distinct from mass moment of inertia) measures how a shape's material is distributed relative to a bending axis, and it directly governs a beam's resistance to bending deflection. For a rectangular section of width b and height h about its centroidal axis, the formula is I = bh³/12 — the cube on the height reflects why standing a beam on edge (tall and narrow) resists bending far more than laying it flat, even with identical cross-sectional area.

This calculator takes a rectangle's width and height and returns its moment of inertia along with the section modulus (S = I/c, where c is the distance to the extreme fiber), which converts directly into bending stress via the flexure formula. Structural and mechanical engineers, and students working through statics or mechanics of materials coursework use these values to size beams and joists, verify that a section can carry a design load, and compare stiffness between different rectangular profiles.

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Comments (1)

  • Editorial team

    Section modulus connects the number to stress: bending moment divided by it gives stress, and a moment in N·mm over a modulus in mm³ comes out in MPa. A 1 kN·m moment on the default section, with the 100 mm side vertical, gives 12 MPa. Turning the section on its side would use Sy instead, which is four times smaller here. Which sections do you work with besides rectangles?

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