L / Angle Section Properties Calculator
Enter the two leg lengths and the thickness of an equal or unequal angle and get its area, centroid, second moments, product of inertia, principal axes, minimum radius of gyration, section moduli, torsion constant and shear centre, 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 angle is the textbook example of a section whose principal axes are rotated: it has no axis of symmetry parallel to its legs, so the product of inertia Ixy is not zero and the true maximum and minimum second moments occur about a pair of axes at an angle θp to the legs — exactly 45° for an equal angle, less for an unequal one. Bending an angle about a leg-parallel axis produces deflection in both directions.
For a single angle used as a strut, what governs buckling is the minimum radius of gyration r2 = √(I2/A) about the minor principal axis, not rx or ry: the calculator reports both principal radii whenever the axes are rotated. The drawing shows the principal axes in green and the shear centre S at the corner where the leg mid-lines meet.
How the l / angle section properties are calculated
- Area
- A = t·(a + b − t)
- Centroid from the outer corner
- x̄ = [b²·t + (a − t)·t²] / (2·A), ȳ = [a²·t + (b − t)·t²] / (2·A)
- Second moments about the centroidal axes
- Ix, Iy by the parallel-axis theorem on the two leg rectangles (vertical leg t × a, horizontal leg (b − t) × t)
- Product of inertia
- Ixy = Σ Ai·(xi − x̄)·(yi − ȳ) over the two rectangles
- Principal angle
- tan 2θp = −2·Ixy / (Ix − Iy)
- Principal moments
- I1,2 = (Ix + Iy)/2 ± √[((Ix − Iy)/2)² + Ixy²]
- Torsion constant (thin-walled)
- J ≈ (a + b − t)·t³ / 3
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 root radius and no toe radius. Rolled angles have slightly larger A and I (1–3%) and a shear centre very close to the leg intersection. The torsion constant is the thin-walled open-section approximation.
Other sections
Frequently asked questions
Why is the principal angle of an equal angle exactly 45°?
Because an equal angle is symmetric about its bisector: a symmetry axis is always a principal axis, and the bisector is at 45° to both legs. The maximum second moment I_1 is about the axis perpendicular to the bisector (through the two toes), the minimum I_2 about the bisector itself, which is the buckling axis of an equal-angle strut.
What is the sign convention of the product of inertia?
I_xy = ∫x·y dA with x and y measured from the centroid, no minus sign — the convention of Hibbeler, Gere and Roark. For an angle with its legs along +x and +y from the corner, most of the area lies in the second and fourth quadrants relative to the centroid, so I_xy comes out negative; the principal angle θp = ½·atan2(−2·I_xy, I_x − I_y) is then measured counter-clockwise from the x axis to the axis of I_1.
Which radius of gyration do I use for an angle strut?
The minimum one, r_2 = √(I_2/A) about the minor principal axis, unless the angle is restrained against buckling in that direction. It is noticeably smaller than r_x or r_y — for a 100 × 65 × 8 angle r_2 ≈ 14.2 mm against r_y ≈ 18.6 mm — so using a leg-parallel radius would over-estimate the buckling load.
How do the results compare with rolled-angle tables?
Area, centroid and I_x / I_y are within a couple of percent (the root radius adds a little material near the corner). Tables quote the principal moments as I_u / I_v and the angle as tan α measured from the long leg to the minor axis — the same angle as θp on this page for an angle with its long leg vertical.
References & further reading
- Second moment of area — Wikipedia — definition, parallel-axis theorem and the sign convention for the product of inertia.
- Radius of gyration — Wikipedia — r = √(I/A) and its role in the slenderness ratio for column buckling.
- Shear centre — Wikipedia — the point a transverse load must pass through to bend a beam without twisting it.
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