The moment of area is a geometric property of a cross-section that describes how its area is distributed relative to an axis. It comes in two forms: the first moment of area (used to find centroids) and the second moment of area (used to predict how a beam resists bending or deflection). The second moment of area is the one most people are looking for, and its value depends entirely on the shape and dimensions of the cross-section. Units are length to the fourth power, typically mm⁴ or in⁴.
First vs. Second Moment of Area
The first moment of area measures how a shape’s area is distributed relative to an axis, and it’s primarily used to locate the centroid of a cross-section. For a shape broken into simple parts, the centroid coordinates are:
- x̄ = (A₁·x̄₁ + A₂·x̄₂ + A₃·x̄₃ + …) / A_total
- ȳ = (A₁·ȳ₁ + A₂·ȳ₂ + A₃·ȳ₃ + …) / A_total
Each term is the area of one sub-shape multiplied by its own centroid distance from a chosen origin, all divided by the total area. You need to find the centroid before you can accurately calculate the second moment of area for composite shapes.
The second moment of area (often called the area moment of inertia) tells you how resistant a cross-section is to bending. A beam with a large second moment of area deflects less under the same load than one with a small value. This is why I-beams are shaped the way they are: pushing material far from the center increases the second moment of area dramatically, even without adding more material. The physical meaning is completely different from the mass moment of inertia used in rotational dynamics, even though they share similar-looking equations.
Formulas for Common Shapes
For standard shapes, the second moment of area about an axis through the centroid has a known formula. In every case below, the axis passes through the shape’s center.
Rectangle
For a rectangle with width b and height h:
- I_x = bh³ / 12 (about the horizontal axis)
- I_y = b³h / 12 (about the vertical axis)
As an example, a rectangle 50 mm wide and 70 mm tall has I_x = (50 × 70³) / 12 = 1.43 × 10⁶ mm⁴. Notice that height is cubed in the horizontal-axis formula, so doubling the height increases bending resistance eightfold while doubling the width only doubles it.
Circle
For a solid circle with radius r (or diameter d):
- I_x = I_y = πr⁴ / 4 = πd⁴ / 64
Because a circle is symmetric, both axes give the same result.
Hollow Circle (Pipe or Tube)
For a hollow circular section with outer radius r_o and inner radius r_i:
- I_x = I_y = π(r_o⁴ − r_i⁴) / 4
You subtract the “missing” inner circle from the solid outer circle. This is why hollow tubes are so efficient: removing material from the center, where it contributes little to bending resistance, saves weight with minimal loss of stiffness.
Triangle
For a triangle with base b and height h, about an axis through its centroid:
- I_x = bh³ / 36
- I_y = b³h / 36
Square
A square with side length b is just a special case of the rectangle formula:
- I_x = I_y = b⁴ / 12
How to Calculate Composite Shapes
Real cross-sections (I-beams, T-beams, channels, built-up sections) are rarely simple rectangles or circles. The standard approach is to divide the complex shape into simpler parts, calculate each part’s contribution, then combine them. Here’s the step-by-step process:
1. Set up a reference point. Label an origin and x/y axes on your shape. All measurements need to be taken from this same point.
2. Break the shape into simple pieces. An I-beam, for instance, becomes three rectangles: a top flange, a web, and a bottom flange. If your shape has holes or cutouts, treat those as negative areas.
3. Find the overall centroid. Use the first moment of area formulas above. Calculate the area of each piece, find each piece’s centroid location relative to your origin, then use the weighted average to find the combined centroid.
4. Apply the parallel axis theorem to each piece. Each sub-shape has its own second moment of area about its own centroid. To shift that value to the combined centroid, you add a correction term:
I_total = I_centroid + A·d²
Here, I_centroid is the second moment of area of that sub-shape about its own center, A is the sub-shape’s area, and d is the distance between the sub-shape’s centroid and the overall centroid. This correction term always adds to the value, so pieces farther from the center contribute more.
5. Sum all contributions. Add the adjusted second moment of area for every solid piece. Subtract the adjusted values for any holes or cutouts. The result is the second moment of area for the entire composite cross-section.
This is exactly the principle behind I-beam design. By placing most of the cross-sectional area in the flanges, far from the centroid, the A·d² term becomes large, and the beam resists bending far more effectively than a solid rectangle of the same total area.
Worked Example: Hollow Rectangle
Suppose you have a box section that is 50 mm wide and 70 mm tall on the outside, with walls 15 mm thick, creating a hollow rectangular tube. The outer rectangle has I_x = (50 × 70³) / 12 = 1.43 × 10⁶ mm⁴. The inner cutout is 20 mm wide and 50 mm tall (after subtracting the wall thickness from each side), giving I_x = (20 × 50³) / 12 = 0.21 × 10⁶ mm⁴.
Because both rectangles share the same centroid, no parallel axis correction is needed. The final value is simply 1.43 × 10⁶ − 0.21 × 10⁶ = 1.22 × 10⁶ mm⁴. For shapes where the sub-parts don’t share a centroid, the parallel axis theorem handles the offset.
Polar Moment of Area
The polar moment of area, J, measures resistance to twisting rather than bending. While the regular (planar) second moment of area predicts how a beam deflects under a load, the polar moment predicts how much a shaft twists under torque. For a solid circular shaft:
J = πD⁴ / 32
For a hollow circular shaft with outer diameter D and inner diameter d:
J = π(D⁴ − d⁴) / 32
The polar moment equals the sum of the two planar second moments of area (J = I_x + I_y), which is why circular sections, where I_x and I_y are equal, are the natural choice for shafts that must resist torsion.
Radius of Gyration
The radius of gyration, r, compresses the second moment of area into a single length value that represents how far from the axis you’d need to concentrate all the area to get the same I value. The formula is:
r = √(I / A)
where I is the second moment of area and A is the total cross-sectional area. This quantity is particularly useful in column buckling analysis, where a smaller radius of gyration about any axis means the column is more likely to buckle in that direction. For non-symmetric shapes, you calculate it separately for each axis, and the smaller value identifies the weakest direction.