Ellipse Calculator
This ellipse calculator takes the equation in any form and returns the centre, a and b, the four vertices, both foci and the eccentricity — with c and e left exact rather than rounded to a decimal.
Type 4x² + 9y² = 36 and you get centre (0, 0), a = 3, b = 2, and foci at c = √5 with eccentricity √5/3. Not 2.236.
What the Ellipse Calculator Is Working From
Pick two points and call them the foci. An ellipse is every point whose two distances to them add up to the same total. That total is 2a, the full length of the long axis.
That one sentence explains the rest of the page. It says why the foci sit inside the curve, why the shape is symmetric about both axes, and why a circle counts as an ellipse — the case where the two foci have met.
Get It to the Divided Form First
Complete the square on x and on y separately, move the constant across, then divide the whole equation by it so the right-hand side becomes 1:
(x − h)²/a² + (y − k)²/b² = 1
Only in that shape can anything be read off. The centre is (h, k), and the two denominators are the squared radii — a and b are their square roots, which is where the first slip usually happens.
a Is the Bigger One, Wherever It Sits
This is the step people get wrong most, because textbooks nearly always draw a under the x term. It is not a rule about position.
a is whichever denominator is larger, and the long axis runs in that direction. In x²/9 + y²/25 = 1 the bigger number is under y, so a = 5, b = 3, and the ellipse is tall. Read the numbers, not the layout.
Subtract for the Foci, Do Not Add
c² = a² − b²
An ellipse subtracts; a hyperbola adds. Swapping the two is the commonest mistake in the whole chapter, and there is a reason worth holding on to rather than a rule to memorise: the foci of an ellipse are inside it, so c has to come out smaller than a — and only subtracting can do that.
Then step c from the centre along the long axis, both ways, and you have the foci.
c and e Are Usually Irrational
a² and b² are whole far more often than their difference is a perfect square. For 4x² + 9y² = 36, a² = 9 and b² = 4, so c² = 5 and c = √5. There is no tidier way to write it.
Round it to 2.236 and the error travels: the foci move, the eccentricity inherits it, and a check against the curve stops coming out equal. This page keeps the surd all the way, and gives the eccentricity the same way — √5/3, not 0.745.
What the Eccentricity Tells You
e = c/a, always between 0 and 1. It describes the shape, not the size: double every measurement and e does not move, which is why astronomers quote it for an orbit.
Near 0 the ellipse is almost round. Near 1 it is a long thin sliver. At exactly 0 the two foci have met at the centre and you have a circle.
A Circle Is an Ellipse
When a and b come out equal, c is zero, both foci sit on top of the centre, and e = 0. Every diameter is then the same length, so there is no long axis to name — and this page says so rather than picking one.
It is worth seeing a circle as a member of the family rather than a different shape. Everything above still holds; the numbers have just collapsed to their simplest case.
When It Is Not an Ellipse
An ellipse needs x² and y² with the same sign. Opposite signs give a hyperbola; only one squared term gives a parabola.
Two other cases are worth naming. If the constant works out so that no point satisfies the equation, there is nothing there at all; if it works out to exactly zero, the whole ellipse has shrunk to a single point. And an equation with an xy term can still be an ellipse — a turned one, with its long axis at an angle. Calling that "not an ellipse" would be a wrong answer rather than a refusal, so it is named for what it is.
Check It on the Curve
Take any point of the ellipse and measure to both foci. The two distances must add to 2a, every time, because that is the definition rather than a consequence of it.
The check costs a minute and catches a wrong c, a swapped a and b, and a sign slip in one go — which is more than re-reading your own working will do.
Using the Ellipse Calculator
Put the equation in the box, however it is written, and tap the button. You get the centre, a, b and c, all four ends of the axes, both foci, the eccentricity, both axis lengths, the area and the latus rectum, a full working that completes the squares line by line, and a drawing with the two distances marked — so you can see the definition rather than take it on trust.
Ellipse Calculator FAQ
How do you find the foci of an ellipse?
Which is a and which is b in an ellipse?
What is the eccentricity of an ellipse?
Is a circle an ellipse?
Why is it a squared minus b squared and not plus?
Divide through first, take a as the larger denominator, subtract for c — and keep the surd to the end.
