How this calculator works
The second moment of area, often called the area moment of inertia, measures how widely a shape spreads its material around an axis. It is the quantity that governs how strongly a beam resists bending. Moving material farther from the axis raises the value quickly because each small area element is multiplied by the square of its distance. The centroidal values are measured about axes through the shape centre; the corner values use two edges of the rectangle. The polar second moment is the sum of the two perpendicular values and describes resistance to twisting about the point where the axes cross. This calculator assumes inputs are converted to metres, so the results are in metres to the fourth power.
Formula
Rectangle about centroid: Ix = bh³/12, Iy = hb³/12. Rectangle about a corner: Ix = bh³/3, Iy = hb³/3. Square: Ix = Iy = a⁴/12. Triangle about its centroid: Ix = bh³/36, Iy = hb³/48. Circle: Ix = Iy = πr⁴/4. Hollow circle: Ix = Iy = π(R⁴ - r⁴)/4. Ellipse with semi-axes a along x and b along y: Ix = πab³/4 and Iy = πba³/4. Polar J = Ix + Iy in every case.
Worked example
A rectangle 4 m wide and 6 m high about its centroid has Ix = 4 × 216/12 = 72 m⁴, Iy = 6 × 64/12 = 32 m⁴, polar moment 104 m⁴ and area 24 m². The same rectangle about a corner has Ix = 288 m⁴ and Iy = 128 m⁴.
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Frequently asked questions
Is this the same as the moment of inertia in physics?
Not quite. This is the second moment of area, used for bending and deflection of beams. Mass moment of inertia also involves density and is used for rotation of a solid body.
Why is the corner value larger?
The parallel axis theorem adds Ad² when the axis is moved away from the centroid. A corner axis is farther from most of the area, so the second moment grows.
Which axis should I use for beam bending?
Use the centroidal axis that matches the direction of bending. A beam bent about its strong axis uses the larger of Ix and Iy for the section orientation.
Why is the result shown without a unit selector?
The site has no fourth-power length unit, so the values are shown in metres to the fourth power after the inputs are converted to metres.