Square Section Properties Calculator
Enter the side of a solid square bar and get its area, second moments, section moduli, radius of gyration and torsion constant, with the section drawn to scale.
Section dimensions
Enter the dimensions in the selected unit — the drawing and the results update as you type. The drawing uses centroidal coordinates: the bold gridlines are the x and y axes through the centroid C.
Section
- Section with its dimensions
- Centroid C, centroidal axes x and y
- Principal axes 1 and 2, angle θp
- Shear centre S
The square bar is the simplest section with a non-trivial torsion constant: unlike the circle, a square does not stay plane when twisted, so J is not the polar moment. Saint-Venant’s solution gives J ≈ 0.1406·a⁴, against Ip = a⁴/6 ≈ 0.1667·a⁴ — using the polar moment would over-estimate the torsional stiffness by about 19%.
In bending, the square is isotropic: every centroidal axis has the same second moment a⁴/12, so the principal axes are undefined and I1 = I2. The section modulus, however, does depend on orientation — bent about a diagonal the extreme fibre is √2 times farther away, so S drops by a factor of √2.
How the square properties are calculated
- Area
- A = a²
- Second moment of area
- Ix = Iy = a⁴ / 12
- Polar moment
- Ip = a⁴ / 6
- Elastic section modulus
- S = a³ / 6
- Plastic section modulus
- Z = a³ / 4
- Radius of gyration
- r = a / √12 ≈ 0.2887·a
- Torsion constant (Saint-Venant)
- J ≈ 0.1406·a⁴
Principal moments and axes follow from Ix, Iy and Ixy with I1,2 = (Ix + Iy)/2 ± √[((Ix − Iy)/2)² + Ixy²] and tan 2θp = −2·Ixy/(Ix − Iy); the radii of gyration are r = √(I/A) and the elastic section moduli S = I/c for each extreme fibre. Definitions of every property are in the glossary on the section properties overview.
Assumptions. Sharp corners, no edge radius. The torsion constant is Roark’s closed-form fit to the Saint-Venant solution, accurate to about 1%.
Other sections
Frequently asked questions
Why is the moment of inertia the same about any axis through the centre?
The square has two perpendicular axes of symmetry with equal second moments, which makes the inertia tensor isotropic in the plane: I_x = I_y and I_xy = 0 for every rotation. The same holds for every regular polygon and for the circle.
Is a square bar stiffer than a round bar of the same width?
Yes. With a = d, I_square = a⁴/12 ≈ 0.0833·a⁴ versus I_circle = πd⁴/64 ≈ 0.0491·d⁴, so the square is about 70% stiffer in bending — but it also has 27% more area. For the same area the two are within 5% of each other.
Where is the maximum torsional shear stress on a square shaft?
At the middle of each side, not at the corners: the corners are stress-free. Roark gives τ_max ≈ 4.81·T/a³ for a solid square, which is why the polar-moment shortcut τ = T·r/I_p is wrong for non-circular shafts.
References & further reading
- List of second moments of area — Wikipedia — closed-form Ix and Iy for the common shapes, used to check this calculator.
- Torsion constant — Wikipedia — why J differs from the polar moment for non-circular sections, with the thin-walled formulas.
- Section modulus — Wikipedia — elastic and plastic section modulus, with a table of formulas by shape.
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