Free Online Cross Section Property Calculator
Welcome to the Cross Section Property Calculator. A free, online cross section calculator to determine moment of inertia, centroids, cross section area and polar moment of inertia. Select a shape and enter dimensions to get started.
| Property | Value |
|---|---|
| A | |
| xc | |
| yc | |
| Ix | |
| Iy | |
| Ixy | |
| Ip |
What the Calculator Gives You
Pick a shape, enter its dimensions in inches or millimeters, and the calculator returns its section properties, with the formula used for each one:
- A, the cross section area.
- xc, yc, the centroid, measured from the origin shown on the shape's drawing: the bottom left corner for most shapes, the center for a circle or pipe.
- Ix, Iy, the moments of inertia (second moments of area) about horizontal and vertical axes through the centroid.
- Ixy, the product of inertia, which is zero for shapes symmetric about either axis.
- Ip, the polar moment of inertia about the centroid, Ix + Iy.
Shapes: rectangle, general, isosceles and right triangles, solid circle (rod), pipe, half circle, and I, T, L and C shapes built from rectangles. The product and polar moments of inertia aren't given for the I, T, L and C shapes.
Example: a rectangle
For a 4 in wide by 12 in deep rectangle:
- A = bh = 4 × 12 = 48 in²
- Ix = bh³/12 = 4 × 12³ / 12 = 576 in⁴
- Iy = hb³/12 = 12 × 4³ / 12 = 64 in⁴
- Ip = Ix + Iy = 640 in⁴
The same board laid flat is nine times less stiff in bending (64 in⁴ against 576 in⁴), which is why joists and beams stand on edge.
What section properties are used for
- Moment of inertia sets how much a beam deflects; the beam calculator uses it with the span and load.
- Moment of inertia divided by the distance from the centroid to the extreme fiber gives the elastic section modulus S, which sets the bending stress M/S.
- The radius of gyration, r = √(I/A), sets how easily a column buckles; see the column designer.
- For standard steel shapes, use the tabulated AISC properties on the steel beam sizes page instead; they include the fillets that this calculator leaves out. The I-beam moment of inertia example shows the difference.
Frequently asked questions
What is the moment of inertia of a rectangle?
About its horizontal centroidal axis, Ix = bh³/12, where b is the width and h the depth.
What is the moment of inertia of a circle?
For a solid circle of radius r, I = πr⁴/4 about any axis through its center, and the polar moment of inertia is πr⁴/2.
Is polar moment of inertia the same as the torsion constant?
Only for solid circles and round tubes. For other shapes, such as I shapes and rectangles, the torsion constant J is much smaller than Ip.