Engineer's Calculator

Cross Section Calculator

Area, centroid, moments of inertia, section modulus and radius of gyration for 11 common shapes. The shape is drawn to scale with its centroid as you type. Free, no sign-up.

1. Pick a shape

Looking for a standard steel section? WebStructural lists the properties of every AISC W shape.

2. Enter the dimensions

3. Section properties

x̄ and ȳ are measured from the left and bottom edges. The moments of inertia are about axes through the centroid. Section modulus is to the farther edge, which gives the larger bending stress.

Formulas for this shape

Checking a beam with this section? Take I to the beam calculator for deflection and moment, or size a steel or wood beam to code in the WebStructural beam designer.

Open the beam designer

What the properties mean

Moment of inertia (I, also called the second moment of area) measures how a shape's area is spread away from an axis. Bending stiffness is E×I, so doubling I halves the deflection.

Section modulus (S = I / c, where c is the distance to the extreme fiber) turns a bending moment into a stress: stress = M / S.

Radius of gyration (r = √(I / A)) is used in column buckling. The smaller one usually governs.

Product of inertia (Ixy) is zero when the shape is symmetric about either axis. When it isn't zero, as for an angle, the shape bends most easily about a tilted principal axis, and the calculator gives the principal moments of inertia and the angle.

Worked example: a 2×10 joist

A 2×10 is actually 1.5 in by 9.25 in. Enter b = 1.5 and h = 9.25 for the rectangle: A = 13.88 in², Ix = 1.5 × 9.25³ / 12 = 98.93 in⁴ and Sx = 21.39 in³, which match the published lumber tables.

Questions

How are the I, T, L and C shapes calculated?

Exactly, from the outline, without the corner fillets of rolled steel shapes. That's why results for a rolled W shape come out a little lower than the steel manual's values.

What is the polar moment of inertia used for?

J = Ix + Iy is the torsion constant for solid and hollow circular shafts. For open shapes like I and C sections the real torsion constant is much smaller, so don't use J for their twisting.