Ellipse area:
formula and worked example

Ovals in the real world

An ellipse is a circle that has been stretched along one direction. Oval flower beds, running tracks drawn as a long oval, elliptical mirrors and table tops, the cross-section of a slanted pipe cut, cam profiles and planetary orbits are all ellipses or close to them. When you need the surface an oval covers, such as paint, turf, glass or polish, you need its area.

What makes the ellipse friendly is that its area has a closed form that is as simple as the circle's. The harder question, its perimeter, has no such neat formula, so it is worth keeping the two apart from the start.

Another place ellipses hide is in perspective. A circle viewed at an angle appears as an ellipse whose long axis is the true diameter and whose short axis is the diameter times the cosine of the viewing angle, so area on screen shrinks by that same cosine.

The formula and what a semi-axis is

Area equals pi times the semimajor axis times the semiminor axis. The semimajor axis is half of the longest span across the ellipse, running through the centre. The semiminor axis is half of the shortest span, at right angles to the first.

Pi times a times b generalises pi times r squared: when a and b are equal the shape is a circle, and the formula collapses to the familiar one. That is a reliable way to remember which numbers go where.

An ellipse with semi-axes 6 and 3

Take an oval whose full length is 12 units and full width is 6 units, so its semi-axes are 6 and 3.

1. 1Identify the semi-axes: 12 / 2 = 6 and 6 / 2 = 3.
2. 2Multiply them: 6 x 3 = 18.
3. 3Multiply by pi: 18 x 3.14159 = 56.549.
4. 4Add the square unit: 56.549 square units.

A sense check uses the bounding rectangle. The rectangle that just contains the ellipse measures 12 by 6, which is 72. An ellipse fills pi/4 of its bounding rectangle, about 78.5%, and 56.549 / 72 is 0.7854. Notice also that the area is exactly half of the circle with radius 6 (113.1), because the minor axis is half the major.

What the area represents

The area tells you the flat surface inside the oval. For an elliptical lawn bed of 4 m length and 2.5 m width, the semi-axes are 2 and 1.25, so the area is pi x 2.5 = 7.85 square metres. With that figure you can order topsoil by depth, compost per square metre or edging separately by perimeter.

Be careful not to use the area to infer how long the boundary is. Two ellipses with the same area can have very different edge lengths, and a long thin oval has a much longer perimeter than a rounder one of the same area. If you need edging, treat that as its own calculation.

It is also worth knowing that area and perimeter are only loosely linked for ovals. Among all shapes of a given area the circle has the shortest edge, so squashing a circle into a thin ellipse of the same area makes the boundary longer and the edging, wall or trim more expensive.

Stretching only the long axis

Going from 6 to 6.6 on the long axis lifts the area from 56.549 to 62.204, up 10%. That is a straightforward proportional change because only one of the two factors moved. If both axes grow by 10%, the area rises by 21%, as for a circle, because the stretch applies in two directions.

This is a quick tool for design trade-offs: widening an oval table by 5 cm on both sides adds area proportional to the existing length, while lengthening it by 5 cm adds area proportional to the existing width. The shorter dimension is the more expensive one to extend per centimetre.

Where ellipse areas go wrong

• Using full axes instead of semi-axes. Pi x 12 x 6 = 226.2, four times the correct 56.5.
• Taking the formula for the circle and substituting the major axis for the radius, which gives 113.1 and ignores the narrower width.
• Assuming the perimeter is pi x (a + b) or 2 x pi x average axis. These are only rough and can be wrong by several per cent for elongated ovals.
• Measuring an oval that is not a true ellipse, such as a racetrack with straight sides and semicircular ends, and treating it as one.

Finally, there is the question of which dimension was measured. A stated length of an oval sign is usually the overall extent, equal to twice the semimajor axis. If a drawing gives a half-length or a radius at the centre, double it before comparing with other figures.

Scenario: glass for an oval window

An oval window frame has an inside length of 90 cm and an inside height of 60 cm. The glass area is pi x 45 x 30 = 4,241.2 square centimetres, or 0.424 square metres. The glazier will cut from a rectangle at least 90 by 60 cm, so you pay for 5,400 square centimetres of rectangular stock and waste about a fifth. Seeing both numbers side by side makes it clear why curved shapes cost more per useful square metre than rectangular ones.