Ellipse Area Calculator
Find the area and circumference of an ellipse.
Results update as you type.
How it works
An ellipse is a stretched circle defined by two axes, and this calculator finds its area and approximate circumference from the semi-major and semi-minor axes. You enter the two semi-axes, half the width and half the height of the ellipse, and it returns the exact area and a very accurate estimate of the perimeter. The area formula is elegantly simple: pi multiplied by the two semi-axes, which generalises the familiar circle area since a circle is just an ellipse whose axes are equal. The circumference, by contrast, cannot be written with elementary functions and requires either an infinite series or an approximation; this tool uses Ramanujan's second approximation, which is accurate to a tiny fraction of a percent for ordinary ellipses. Ellipses appear throughout the real world: planetary orbits are ellipses, as are many arches, running tracks, gears, and design elements. Engineers, architects, astronomers, and students all need to compute their area and perimeter. Remember that the inputs are the semi-axes, not the full axis lengths, so enter half of the total width and height. Enter the semi-major and semi-minor axes to compute the area and circumference of your ellipse.
Area = π·a·b
The area of an ellipse is pi times the product of its two semi-axes, the semi-major axis a and the semi-minor axis b. When a equals b the ellipse becomes a circle and the formula reduces to pi times the radius squared. The circumference has no simple closed form, so this calculator uses Ramanujan's accurate approximation.
Source:Euclidean geometry
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
Area updates instantly as you type — no button to press.
Frequently asked questions
How do you find the area of an ellipse?
The area of an ellipse equals pi times the semi-major axis times the semi-minor axis, where the semi-axes are half the full width and half the full height. For an ellipse with semi-axes of three and two, the area is six pi, about 18.85 square units. This calculator applies the formula automatically when you enter the two semi-axes.
What is the difference between the axes and semi-axes?
The major and minor axes are the full width and height of the ellipse across its centre, while the semi-major and semi-minor axes are exactly half of each. The area formula uses the semi-axes. If you know the full axis lengths, divide each by two before entering them, or your result will be four times too large.
Why is the circumference only approximate?
Unlike its area, the exact circumference of an ellipse cannot be expressed with elementary functions; it requires a complete elliptic integral or an infinite series. This calculator uses Ramanujan's second approximation, which is remarkably accurate, typically within a small fraction of a percent for ordinary ellipses, and is far more precise than simpler estimates.
Is a circle a special ellipse?
Yes. A circle is an ellipse in which the semi-major and semi-minor axes are equal, both equal to the radius. In that case the area formula pi times a times b becomes pi times the radius squared, the standard circle area, and the circumference approximation returns two pi times the radius, the exact circle circumference.
What are ellipses used for?
Ellipses model planetary and satellite orbits, which are ellipses with the central body at a focus, and they appear in optics, antenna design, architecture such as elliptical arches and domes, running tracks, and graphic design. Calculating their area and perimeter is essential in engineering, astronomy, manufacturing, and any field involving oval shapes.