Second Moment of Area Triangle Calculator

This calculator computes the second moment of area (area moment of inertia) of a triangle about a chosen axis, given the triangle’s base width, height, and the axis location. Enter the base and height, select the axis (centroidal or base), and get the result in mm�, cm�, or in�—the tool handles the unit conversion for you.

Formula

The second moment of area (often called the area moment of inertia) measures how a cross-section’s area is distributed relative to an axis. It is not the same as mass moment of inertia—it has units of length� (e.g., mm�, in�).

For a triangle with base b and height h:

• About the centroidal axis (parallel to the base): I_c = (b × h³) / 36
• About the base axis (the bottom edge): I_b = (b × h³) / 12

Where:

• b = base width (length)
• h = height perpendicular to the base (length)
• I = second moment of area (length�)

These formulas assume a solid, homogeneous triangle with a constant thickness (i.e., a 2D cross-section). The centroidal axis is located at a distance h/3 from the base.

Worked Example

Using the default values: base = 100 mm, height = 150 mm, axis = centroidal.

1. Calculate h³: 150³ = 3,375,000 mm³
2. Multiply by base: 100 × 3,375,000 = 337,500,000 mm�
3. Divide by 36: 337,500,000 / 36 = 9,375,000 mm�

The calculator shows 9,375,000 mm� (or 9.375 × 10� mm�). If you switch to the base axis, the result becomes 28,125,000 mm� (three times larger).

How to Read the Result

The second moment of area indicates stiffness against bending. A larger value means the cross-section resists bending more. For a triangle, the centroidal axis gives a smaller value than the base axis because the area is closer to the centroidal axis on average. When you compare different cross-sections (e.g., a rectangle vs. a triangle of the same base and height), the rectangle has I = b·h³/12 about its centroid, which is three times larger than the triangle’s centroidal value—so a rectangular beam is stiffer in bending.

In structural engineering, this value feeds directly into beam deflection and stress formulas (e.g., σ = M·c/I). Always match the axis you use in your calculations to the axis in the beam formula.

When to Use This Calculator

• Designing a triangular cross-section beam or bracket.
• Calculating deflection of a triangular plate or fin.
• Checking stress in a triangular structural member under bending.
• Converting between units (mm� to in�, etc.) for comparison.

This is a pure geometry calculation—it does not depend on material. The same formula works for steel, aluminum, wood, or plastic.

Common Mistakes and Misconceptions

• Using the wrong axis: The most common error. The centroidal formula is (b·h³)/36, the base formula is (b·h³)/12. Mixing them changes the result by a factor of 3.
• Forgetting to cube the height: The height is cubed, not the base. Doubling the height multiplies the moment by 8, not 2.
• Unit mix-ups: If you input in mm, the result is in mm�. To get cm�, divide by 10,000 (since 1 cm = 10 mm, so 1 cm� = 10� mm�). To get in�, divide by 25.4� ≈ 416,231.
• Confusing with mass moment of inertia: This calculator gives area moment (length�), not mass moment (mass·length²).

Limitations of the Formula

The formula assumes a perfectly straight, two-dimensional triangle with uniform thickness. It does not account for holes, fillets, or non-uniform material. For a triangle with a different orientation (e.g., rotated), you would need the product of inertia or use the parallel axis theorem. Also, the formula gives the moment about an axis parallel to the base—if you need the moment about a vertical axis, you must use a different formula (I_y = (h·b³)/48 for centroidal, for example).

Frequently Asked Questions

What is the difference between centroidal and base axis?

The centroidal axis passes through the centroid (geometric center) of the triangle, located at one-third of the height from the base. The base axis lies along the bottom edge. The moment about the base is exactly three times larger than about the centroid because of the parallel axis theorem: I_base = I_centroid + A·d², where d = h/3 and A = (b·h)/2.

Can I use this for a right triangle?

Yes, the formula works for any triangle shape—right, isosceles, or scalene—as long as you use the base and the perpendicular height. The base does not have to be the longest side; it is simply the side you choose as the reference.

How do I convert mm� to in�?

Divide the mm� value by 25.4� (which is approximately 416,231.43). For example, 9,375,000 mm� ÷ 416,231 ≈ 22.53 in�.

Why is the height cubed but not the base?

Because the second moment of area is an integral of y² dA. The height direction (y) appears squared in the integral, and the height also appears in the area element (dA = b·dy), giving a total of h³. The base appears only once, so it is not cubed.

What if my triangle has a hole or is not solid?

This calculator assumes a solid triangle. For a composite shape, calculate the moment of the outer triangle and subtract the moment of the hole (using the same formula for the hole, but with its own dimensions and axis). This is valid only if the hole’s centroidal axis coincides with the outer triangle’s axis—otherwise, use the parallel axis theorem.

Is this the same as the polar moment of inertia?

No. The polar moment of inertia (J) measures resistance to torsion and is the sum of the second moments about two perpendicular axes (J = I_x + I_y). This calculator gives only the moment about one axis. For torsion of a triangular shaft, you would need the polar moment, which is more complex.

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in b | Base width | mm | 100 | 0
in h | Height | mm | 150 | 0
sel axis | Axis | Centroidal=1, Base=2 | 1
out I | Second moment of area | mm� | (b * h^3) / 36 * (axis == 1 ? 1 : 3) | 0