Section Properties Calculator: Area, Second Moment and Section Modulus

Area, second moment of area, section modulus and radius of gyration for common solid and hollow sections.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

The shape of a cross-section decides how well it resists bending. The second moment of area I measures that resistance, the section modulus Z = I / y converts a bending moment into stress, and the radius of gyration r tells you how a column buckles.

This calculator gives the standard values about the centroidal x axis for a solid rectangle, a solid circle, a hollow rectangle and a hollow circle.

Formulas used

  • I_x = \frac{b h^3}{12}\ \text{(rectangle)}
  • I = \frac{\pi d^4}{64}\ \text{(circle)}
  • Z = I / y_{max}

Worked example

With the default numbers in the calculator above:

  1. Area: 15000 mm²
  2. Second moment of area: 28125000 mm⁴
  3. Elastic section modulus: 375000 mm³
  4. Radius of gyration: 43.3 mm
  5. Second moment of area about the other axis (y): 12500000 mm⁴
  6. First moment of area of half the section about the neutral axis: 281250 mm³

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

Why is a deep section stiffer than a wide one?

Because I grows with the cube of the depth. Doubling the depth makes the section eight times stiffer in bending.

Are the hollow shapes with rounded corners?

Yes. Square and rectangular hollow sections use rounded corners for the area, second moments and section modulus. Type in the outside corner radius from the supplier or standard section table for the exact section, or use the estimate from the wall thickness if you do not have the table. You can also choose sharp corners to see the theory case.

Which axis is the x axis?

The horizontal axis through the centroid, so the depth is measured vertically.

Related calculators

Further reading: Hibbeler, Mechanics of Materials; Gere and Goodno, Mechanics of Materials; Shigley, Mechanical Engineering Design.

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