How to use the section properties calculator
- List the rectangles that make up your cross section, one per line, as name, width, height, x, y, where x and y give the lower left corner. Use any one length unit.
- Add the word hole at the end of a line to cut a rectangle out of the shape.
- Type a short unit label, such as in or mm. It is shown after the results and does not change any number.
- Read the area, centroid, moments of inertia, section moduli and radii of gyration, and check the table of pieces.
Formula
area = sum of b h centroid x = sum of (A x_c) / area, y = sum of (A y_c) / area
Ix = sum of (b h^3 / 12 + A d_y^2) Iy = sum of (h b^3 / 12 + A d_x^2)
section modulus = I / (distance from the centroid to the farthest edge) radius of gyration = square root of (I / A)
Each rectangle has its own moment of inertia about its own center, b h cubed over 12 for the horizontal axis. The parallel axis theorem adds area times the square of the distance from that center to the centroid of the whole shape. A hole counts as negative area and negative inertia.
A T-shaped section
A flange 6 wide and 1 high with its lower left corner at (0, 5), joined to a web 1 wide and 5 high at (2.5, 0), has an area of 11 and a centroid at x = 3 and y = 4.136364 (measured from the bottom). Ix is 35.46212, Iy is 18.41667, and the section modulus for the top edge is 19.0285 against 8.5733 for the bottom, because the centroid sits near the flange. The radius of gyration is 1.7955 about the horizontal axis.
A plate 4 wide and 6 high with a hole 2 by 2 whose corner is at (1, 2) has an area of 20, a centroid at y = 3 and Ix of 70.66667. A numerical integration on a fine grid gives the same values to within 0.2%.
Why these numbers matter
The second moment of area, often called the moment of inertia of the section, measures how well a shape resists bending about an axis. It grows with the cube of the depth, which is why a deep narrow beam is much stiffer than a wide shallow one of the same area. The section modulus divides it by the distance to the outer edge and links bending moment to stress at that edge.
The centroid is the balance point of the shape. The bending axis passes through it, and the two section moduli differ whenever the shape is not symmetric top to bottom.
Limits
The page describes geometry only. It does not check strength, stability, connections or loads, and it is not a substitute for a design by a qualified engineer. Pieces of the same kind must not overlap, so merge or split rectangles that do. For the deflection of a beam with a given section, use the Beam Deflection Calculator; for the weight of the metal, the Metal Weight Calculator.
More engineering tools
For wire runs, bends and orbits, try the Sheet Metal Bend Allowance Calculator, Orbital Mechanics Calculator and Wire Voltage Drop Calculator.
Assumptions and limits
- The shape is a union of rectangles with sides parallel to the x and y axes.
- Rectangles of the same kind may touch but may not overlap; holes should lie inside the solid area.
- All dimensions use one length unit, and the second moments are in that unit to the fourth power.
- The shape is a flat cross section with uniform material.
- The example shapes are an illustration. Replace them with your own.
Frequently asked questions
What is the parallel axis theorem?
It says the moment of inertia about any axis equals the moment about a parallel axis through the centroid plus the area times the square of the distance between the axes. It lets you add up pieces that are not centered on the shape.
Why are there two section moduli for each axis?
If the centroid is closer to one edge, the farthest edge on one side is further away than on the other, so dividing the same inertia by different distances gives different moduli.
Can I describe a circle or an I-beam?
An I-beam is three rectangles. A circle is not a rectangle, so it cannot be described here. Approximate shapes with many rectangles only if an approximation suits your purpose.
What does a hole do to the centroid?
It removes area, so the centroid moves away from the hole. The page subtracts the hole's area and its inertia and recomputes the centroid.