Area Of An Ellipse Calculator

What this calculator does

An ellipse is a circle that has been stretched, an oval. Instead of one radius it has two: the semi-major axis, the distance from the centre to the edge the long way, and the semi-minor axis, the same thing the short way.

Give the tool those two half-widths and it returns the area inside the oval.

Using the calculator

1. Type the semi-major axis (a) and pick its unit.
2. Type the semi-minor axis (b) and pick its unit.
3. Press Calculate.

The area comes back in square units of the semi-major axis's unit. The two axes can even be in different units, since the tool lines them up before multiplying.

The formula | area = π × a × b

The area of an ellipse is:

area = π × a × b

where a is the semi-major axis and b is the semi-minor axis, the two half-widths from the centre.

It is the circle's own formula, stretched. A circle's area is π × radius × radius, the same radius used twice. An ellipse simply has two different radii, a one way and b the other, so you swap the second one in and get π × a × b. And unlike the distance around an ellipse, which has no tidy formula, the area is exact. No approximation, no catch.

Where the circle hides inside it

Here is a nice way to remember the formula. A circle is just an ellipse whose two axes happen to be equal. Set a and b both to the same r, and π × a × b becomes π × r × r, which is π × r², the area of a circle. So the ellipse formula is the more general one, and the circle is the special, perfectly balanced case of it.

Units, full axes, and rounding

Each axis carries a unit, and the area lands in the matching square unit, since you are multiplying a length by a length. If you enter a and b in different units, the tool converts one to match the other before working out the area. A tidy result is shown in full, and a long decimal is rounded to four places.

One thing to watch: the tool wants the half-widths, a and b, not the full widths across the oval. If you measured the full long and short axes, halve each one first (a = long axis ÷ 2, b = short axis ÷ 2).

A worked example | a = 5 cm, b = 3 cm

Say the semi-major axis is 5 cm and the semi-minor axis is 3 cm.

1. Multiply the two axes by π: π × 5 × 3 = 15π.
2. Work that out: 15π = 47.1239 cm².

So the area of the oval is about 47.12 cm². For comparison, a circle with a 5 cm radius would be π × 25 = 78.54 cm², larger, because squashing the circle down to a 3 cm short axis takes area away.

Questions people ask

What is the area of an ellipse with a = 5 and b = 3?

It is 15π, about 47.12 square units. Multiply the two half-axes together and by π: π × 5 × 3.

What are the semi-major and semi-minor axes?

They are the ellipse's two radii. The semi-major axis (a) runs from the centre to the edge the long way, and the semi-minor axis (b) runs the short way. Both are measured from the centre.

Is the ellipse area formula exact?

Yes. π × a × b is exact, with no approximation. That is different from the ellipse's circumference, which has no exact elementary formula and has to be estimated.

I measured the full widths, not the half-widths. What do I do?

Halve each one first. The semi-major axis is the long width divided by 2, and the semi-minor axis is the short width divided by 2.

Why is the answer in square units?

Because area is a length times a length. Two axes in centimetres give an area in square centimetres.

References

A note on where this comes from. The ellipse was first studied as one of the conic sections by Apollonius of Perga in his Conics over two thousand years ago, and its area, π × a × b, follows from treating the oval as a stretched circle. The value of π behind it is the one tabulated by the US National Institute of Standards and Technology. If you also need the distance around the oval, see the ellipse circumference calculator. For further reading, see Ellipse.

1. Apollonius of Perga, Conics (c. 200 BCE), the founding study of the ellipse and the other conic sections.
2. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π and properties of the ellipse. https://dlmf.nist.gov/