Ellipse Perimeter Calculator
Estimate an ellipse's perimeter and area.
Results update as you type.
How it works
Unlike a circle, an ellipse has no simple closed-form perimeter, so this calculator uses a highly accurate approximation to estimate it. You enter the semi-major axis and the semi-minor axis, the two half-widths of the ellipse, and it returns the estimated perimeter and the exact area. The perimeter comes from Ramanujan's second approximation, a remarkably precise formula that is accurate to within a tiny fraction of a percent for almost all ellipses, becoming exact for a circle. It depends on a quantity h that captures how stretched the ellipse is: for a circle, where the two axes are equal, h is zero and the formula reduces to the circle circumference two pi r. The area, by contrast, has a simple exact formula: pi times the two semi-axes multiplied together. Ellipses appear throughout science and engineering, from planetary orbits, which are ellipses with the sun at a focus, to gears, lenses, arches, and design elements. Being able to find the perimeter and area lets you calculate material for an elliptical track, the trim around an oval window, or the distance around an orbit. The calculator requires both axes to be positive. When they are equal it correctly returns the circle's circumference and area. Enter the semi-major and semi-minor axes to estimate the ellipse's perimeter and area.
P ≈ π(a+b)(1 + 3h / (10 + √(4−3h))), h = (a−b)²/(a+b)²
An ellipse's perimeter has no simple exact formula, so this uses Ramanujan's second approximation, which is extremely accurate for typical ellipses. It depends on h, a measure of how elongated the ellipse is. The area is exact: π times the semi-major axis times the semi-minor axis.
Source:Geometry (Ramanujan approximation)
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter semi-major axis (a) and semi-minor axis (b).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Perimeter (approx.) updates instantly as you type — no button to press.
Frequently asked questions
Why is there no exact formula for ellipse perimeter?
The exact perimeter of an ellipse is given by a complete elliptic integral, which cannot be expressed with elementary functions. Because of this, mathematicians use approximations. Ramanujan's formulas are among the best, giving results accurate to a tiny fraction of a percent for typical ellipses, which is more than enough for practical purposes.
How accurate is Ramanujan's approximation?
Ramanujan's second approximation, used here, is extraordinarily accurate, with errors far below one percent even for quite elongated ellipses and effectively zero for shapes close to a circle. For nearly all real-world applications the small error is negligible, making it a reliable choice for estimating an ellipse's perimeter without complex integration.
What are the semi-major and semi-minor axes?
The semi-major axis is half the length of the longest diameter of the ellipse, and the semi-minor axis is half the shortest diameter. They are the two radii of the ellipse along its principal directions. When both are equal, the ellipse is a circle, and each becomes the circle's radius.
How is the area of an ellipse calculated?
The area of an ellipse has a simple exact formula: pi times the semi-major axis times the semi-minor axis. This mirrors the circle's area of pi times radius squared, since a circle is an ellipse with equal axes. Unlike the perimeter, the area needs no approximation and the calculator reports it exactly.
Where are ellipse calculations used?
Ellipses model planetary and satellite orbits, which follow elliptical paths, as well as gears, camshafts, lenses, whispering galleries, arches, sports tracks, and many design shapes. Calculating perimeter and area helps with tasks like estimating the material to edge an oval table, the fencing around an elliptical garden, or the length of an orbital path.