Regular Hexagon Section Properties Calculator
Enter the across-flats size of a hexagonal bar and get its area, second moments, section moduli, radii of gyration and across-corners dimension, 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
Hexagonal bar is the raw material of nuts, bolts heads, fittings and turned parts that need a spanner flat, and it is specified by the across-flats dimension s — the wrench size. The across-corners size e = 2s/√3 ≈ 1.155·s is what matters for the clearance of the bar in a collet or the minimum round it can be turned from.
As a regular polygon, the hexagon is isotropic in bending: Ix = Iy = 5√3·a⁴/16 with a = s/√3 the side length, Ixy = 0 for every orientation, and the principal axes are undefined. The section modulus, however, is not the same for all orientations — it is larger when bending about an axis through two flats (extreme fibre at s/2) than through two corners (extreme fibre at e/2).
How the regular hexagon properties are calculated
- Side length and across corners
- a = s / √3, e = 2·a = 2·s / √3
- Area
- A = (3√3 / 2)·a² = (√3 / 2)·s² ≈ 0.8660·s²
- Second moment of area (any centroidal axis)
- I = (5√3 / 16)·a⁴ = (5√3 / 144)·s⁴ ≈ 0.06014·s⁴
- Elastic section modulus, axis through the flats (x)
- Sx = I / (s / 2)
- Elastic section modulus, axis through the corners (y)
- Sy = I / a
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. Regular hexagon with two flats horizontal (corners on the x axis). No torsion constant is given for this shape.
Other sections
Frequently asked questions
Is a hexagonal bar as stiff as the round bar it is turned from?
Stiffer, because it contains more material: a hexagon of across-flats s fits inside a circle of diameter e = 1.155·s and has about 10% more area than a circle of diameter s, with I ≈ 0.0601·s⁴ against 0.0491·s⁴ for the inscribed circle. Compared with the circumscribed circle (diameter e), it has 83% of the area and 66% of the I.
Why is the moment of inertia the same about every axis?
Any regular polygon with three or more sides has an isotropic inertia tensor in the plane: the two principal moments are equal, so I_x = I_y for every rotation and I_xy is always zero. Only the distance to the extreme fibre changes with orientation, which is why S_x and S_y differ.
Which dimension does the calculator ask for?
The across-flats size s, the same number stamped on the bar and matching the spanner size. The across-corners dimension e is shown on the drawing and can be read from it; enter e·√3/2 if all you have is the corner-to-corner size.
References & further reading
- List of second moments of area — Wikipedia — closed-form Ix and Iy for the common shapes, used to check this calculator.
- Section modulus — Wikipedia — elastic and plastic section modulus, with a table of formulas by shape.
- Radius of gyration — Wikipedia — r = √(I/A) and its role in the slenderness ratio for column buckling.
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