Ellipse Section Properties Calculator
Enter the two semi-axes of an elliptical section and get its area, second moments, section moduli, radii of gyration and torsion constant, 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
Elliptical sections appear in cams, lightweight tubing, aerodynamic struts and wherever a round bar is squashed or a bore cuts a cylinder at an angle. The formulas are the circle’s with a and b in place of r: A = πab, Ix = πab³/4 (about the axis parallel to a), and the section is isotropic only when a = b.
The ellipse is also one of the few non-circular shapes with an exact, closed-form torsion constant, J = πa³b³/(a² + b²), which makes it a useful check on approximate methods: for a = b it reduces to the circle’s πr⁴/2, and for a slender ellipse it tends to the thin-strip value.
How the ellipse properties are calculated
- Area
- A = π·a·b
- Second moments of area
- Ix = π·a·b³ / 4, Iy = π·a³·b / 4
- Elastic section moduli
- Sx = π·a·b² / 4, Sy = π·a²·b / 4
- Plastic section moduli
- Zx = 4·a·b² / 3, Zy = 4·a²·b / 3
- Radii of gyration
- rx = b / 2, ry = a / 2
- Torsion constant (exact)
- J = π·a³·b³ / (a² + b²)
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. Exact for a true ellipse with semi-axes a (along x) and b (along y). Enter semi-axes, not full width and height.
Other sections
Frequently asked questions
Do I enter the full axes or the semi-axes?
The semi-axes — half the overall width (a) and half the overall height (b), the same quantities that appear in the textbook formulas. The drawing shows a and b measured from the centre so it is clear which is which.
Is the polar moment of an ellipse equal to its torsion constant?
No, except for the circle. I_p = I_x + I_y = πab(a² + b²)/4, while the exact torsion constant is J = πa³b³/(a² + b²). For a = 2b, J is about 64% of I_p: using the polar moment would over-estimate the torsional stiffness by more than half.
How does an ellipse compare with a rectangle of the same overall size?
The ellipse has π/4 ≈ 78.5% of the rectangle’s area and 3π/16 ≈ 58.9% of its second moment (πab³/4 against 2a·(2b)³/12 = 4ab³/3). Its shape factor Z/S = 16/(3π) ≈ 1.70 is higher than the rectangle’s 1.5.
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.
Carry this calculator in your pocket
MechaHandbook has thread charts, tolerances, standard components, tightening torque and unit conversions — all offline, no internet required.
Download the app