Ellipse Area Calculator
Calculate the area of an ellipse from its semi-major and semi-minor axes.
About
Ellipse Area Calculator
The area of an ellipse is A = πab, where a and b are the two semi-axes: half the long width and half the short width. Enter both above and the calculator shows the working and the exact multiple of π.
What the formula means
An ellipse is a circle that has been stretched along one direction. Stretching a shape by a factor in one direction multiplies its area by that factor, so an ellipse is a circle of radius b stretched by a/b, giving πb² × a/b = πab.
That also explains why the circle formula is the special case. When a and b are equal, both are the radius, and πab becomes πr².
The semi-major axis, a, is half the longest diameter. The semi-minor axis, b, is half the shortest. Using the full widths instead of the halves multiplies the answer by four, and it is the most common error on this page.
A worked example
Take a semi-major axis of 30 units and a semi-minor axis of 20.
- Multiply the two semi-axes: 30 × 20 = 600
- Multiply by π: 600π
- As a decimal: 600 × 3.14159265 = 1884.9556 square units
The check is that the answer must fall between the inscribed and circumscribed circles. A circle of radius 20 covers 1256.6 and one of radius 30 covers 2827.4, and 1884.96 sits between them.
Perimeter has no simple formula
Area is straightforward. Perimeter is not, and this surprises people. There is no exact expression for the circumference of an ellipse in terms of elementary functions. It requires an elliptic integral, which is where that whole family of functions gets its name.
Ramanujan gave an approximation that is accurate enough for almost any practical purpose: P ≈ π[3(a+b) − √((3a+b)(a+3b))]. For a = 30 and b = 20 that gives about 158.65, with an error far below a tenth of a percent.
The crude version, π√(2(a²+b²)), is easier to remember but drifts badly for elongated ellipses. If you are cutting material to length round an elliptical opening, use the Ramanujan form.
Eccentricity, and how squashed an ellipse is
Eccentricity measures the departure from a circle. It runs from 0, a perfect circle, toward 1, a shape flattened almost to a line, and is given by e = √(1 − b²/a²).
| Object | Eccentricity |
|---|---|
| Earth's orbit | 0.0167 |
| Mars's orbit | 0.0934 |
| Halley's Comet | 0.967 |
| Our 30 by 20 example | 0.745 |
Earth's orbit is often drawn as a squashed oval, but at an eccentricity of 0.0167 it is visually indistinguishable from a circle. The seasons come from axial tilt, not from that tiny ellipticity.
The two foci
Every ellipse has two focal points, positioned on the long axis at a distance √(a² − b²) from the centre. The defining property is that the distances from the two foci to any point on the curve always add to the same total, 2a.
That property is the basis of the gardener's method for drawing one: push two pins in, loop a string round them, and trace with a pencil held taut. It also explains whispering galleries, where sound leaving one focus reflects off the curved wall and converges on the other, and lithotripsy, where a shock wave is focused on a kidney stone the same way.
Where it gets used
Orbits. Kepler's first law puts every planet on an ellipse with the Sun at one focus. Orbital areas matter because the second law says equal areas are swept in equal times.
Ducting and tanks. Oval sections fit where a circle will not, and their cross-sectional area sets the flow capacity.
Optics and acoustics. Elliptical reflectors concentrate energy from one focus onto the other.
Engineering sections. Elliptical heads on pressure vessels are shallower than hemispherical ones and use less space for the same containment.
Elliptical rooms, tables and beds
Oval furniture is specified by its two full widths, so halve both before using the formula. A dining table 2.4 m long and 1.1 m wide has semi-axes of 1.2 and 0.55, giving an area of π × 0.66 = 2.07 m².
That figure matters for seating. Allowing 0.6 m of edge per person, the perimeter decides how many fit rather than the area. Using the Ramanujan approximation, this table has a perimeter near 5.69 m, so it seats nine comfortably.
Elliptical ducts and pipes
Flat oval ducting is used where a round duct will not fit in a ceiling void. Its cross-sectional area sets the airflow, and matching a round duct means matching the area rather than the width.
A 300 mm round duct has an area of π × 150² = 70,686 mm². To carry the same flow in a duct only 200 mm deep, the semi-axes must satisfy πab = 70,686 with b = 100, giving a = 225 mm, so a 450 by 200 mm oval. Reading across from width alone would undersize it badly.
Drawing and setting out an ellipse
The string and two pins method needs the focal distance, c = √(a² − b²). For semi-axes of 30 and 20 that is √(900 − 400) = 22.36 units either side of centre, and the string loop must be 2a = 60 units long between the pins.
Gardeners use this to set out oval beds, and stonemasons used it for arches long before the algebra was written down. The method works because the sum of the distances to the two foci is constant, which is the definition of the curve rather than a property of it.
Ellipses from circles: the shadow test
Tilt a circle away from your eye and it appears as an ellipse. The semi-minor axis shortens by the cosine of the tilt angle while the semi-major axis stays put, so the apparent area falls by that same cosine.
A circular manhole cover 600 mm across, photographed at 60 degrees from square on, appears as an ellipse with semi-axes of 300 and 300 × cos 60° = 150 mm. Its apparent area is π × 300 × 150 = 141,372 mm², exactly half the true 282,743 mm².
Surveyors and photogrammetrists reverse this to recover the true shape from a photograph, and astronomers use it to work out the inclination of a galaxy or a ring system from how elliptical it appears.
Comparing an ellipse with its bounding shapes
An ellipse always fills exactly π/4 of the rectangle that encloses it, which is 78.54%, the same fraction a circle fills of its square. The proportion does not depend on how elongated the shape is.
For semi-axes of 30 and 20, the bounding rectangle is 60 by 40 and covers 2400. The ellipse covers 1884.96, and 1884.96 / 2400 is 0.7854. That ratio is a quick check on any elliptical area, and it catches the full-axes error immediately, since that mistake produces an answer larger than the box around it.
Common mistakes
Using full axes instead of semi-axes. This quadruples the answer. The formula needs the halves.
Applying a circle perimeter formula. There is no exact elementary formula for an ellipse perimeter. Use the Ramanujan approximation.
Confusing the semi-minor axis with the distance to a focus. They are different measurements, related by c = √(a² − b²).
Assuming an oval is an ellipse. Many oval shapes, including stadium and egg shapes, are not ellipses and need different formulas.
Common questions
Frequently asked questions
A = pi times a times b, where a and b are the semi-major and semi-minor axes. An ellipse with semi-axes of 30 and 20 has an area of 600 pi, which is 1884.9556 square units.
Half the longest width and half the shortest width. Using the full widths instead of the halves multiplies the answer by four.
A circle is an ellipse whose axes are equal. When a and b are both the radius, pi a b becomes pi r squared.
There is no exact elementary formula. The Ramanujan approximation, pi times 3(a+b) minus the square root of (3a+b)(a+3b), is accurate to well under a tenth of a percent.
A measure of how far from circular an ellipse is, from 0 for a circle toward 1 for a flattened line. It equals the square root of 1 minus b squared over a squared.
Push two pins in at the foci, loop a string around them, and trace with a pencil held taut. The distances to the two foci always add to the same total.
No. Its eccentricity is 0.0167, close enough to a circle to be indistinguishable by eye. The seasons come from axial tilt, not from the orbit shape.
No. An ellipse is a specific curve defined by two foci. Stadium shapes, egg shapes and many drawn ovals are not ellipses and need different formulas.