Ellipse Properties
Enter the center (h, k) and the two semi-axis lengths a (x direction) and b (y direction). The calculator detects whether the major axis is horizontal or vertical, then reports the vertices, co-vertices, foci, eccentricity and area.
Orientation decided, then everything measured from it
An ellipse is the set of points whose distances to two fixed points, the foci, add to a constant. In standard form it is described by four numbers: the centre and the two semi-axis lengths. Everything else — vertices, foci, elongation, enclosed area — follows from those four.
This calculator takes the centre coordinates and the two semi-axes, works out which direction the major axis runs, and reports the vertices, co-vertices, foci, eccentricity and area.
Almost every derived quantity depends on which semi-axis is the longer one. The vertices lie along the major axis and the co-vertices along the minor; the foci sit on the major axis; and the eccentricity divides by the semi-major length, not by whichever one happens to be called a.
Getting that assignment wrong is the classic error, and it silently produces plausible results. The tool decides orientation from the two lengths and states it on its own line, so the assumption behind every later figure is visible first.
How to use this calculator
- Enter the centre as h and k The coordinates of the ellipse's centre. Both may be negative or zero, and the shape is unaffected by them — they only translate it.
- Enter a, the semi-axis in the x direction The distance from the centre to the curve measured horizontally. It must be positive; zero or negative values are rejected rather than interpreted.
- Enter b, the semi-axis in the y direction The corresponding vertical distance, also strictly positive. Whether it exceeds a decides the orientation.
- Read the orientation line before the rest It states whether the major axis is horizontal or vertical. Every subsequent line — vertices, foci, eccentricity — is expressed relative to that choice.
The formula, and where it comes from
(x − h)²/a² + (y − k)²/b² = 1 c = √(a_major² − b_minor²) e = c/a_major A = π·a·b
The standard-form equation says the horizontal and vertical displacements from the centre, each scaled by their own semi-axis, satisfy a Pythagorean-looking identity. Setting a equal to b turns it into the equation of a circle.
The focal distance c is derived from the two semi-axes, always subtracting the smaller squared from the larger squared so the quantity under the root is non-negative. That relation places the foci: they sit on the major axis, c from the centre in each direction, and distances from any point on the curve to the two of them sum to twice the semi-major length.
Eccentricity is that focal distance as a fraction of the semi-major axis, so it always falls between zero and one. Zero means the foci coincide with the centre and the ellipse is a circle; values approaching one mean the foci are pushed almost to the vertices and the curve is nearly flat.
The area formula is π·a·b regardless of which semi-axis is longer, since multiplication is symmetric. It reduces to πr² when the two are equal, which is the sense in which the circle is a degenerate ellipse rather than a different kind of object.
What each input means
- h Centre x-coordinate — form field “Center x (h)”
- Horizontal position of the centre. Any real number; it translates the ellipse without changing its shape.
- k Centre y-coordinate — form field “Center y (k)”
- Vertical position of the centre, with the same role.
- a Semi-axis in x — form field “Semi-axis a (x direction)”
- Distance from the centre to the curve along the horizontal direction. Must be strictly positive. It is the semi-major axis only when it exceeds b.
- b Semi-axis in y — form field “Semi-axis b (y direction)”
- The vertical counterpart, also strictly positive. Note that a and b here name directions, not sizes — either can be the larger.
- c Focal distance
- Not entered — computed as the square root of the difference of the squared semi-axes, and reported with that subtraction shown.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A horizontal ellipse
Centred at the origin with a = 5 and b = 3. The horizontal semi-axis is the longer one, so the major axis runs along x.
Inputs Center x (h) = 0, Center y (k) = 0, Semi-axis a (x direction) = 5, Semi-axis b (y direction) = 3
- Equation (x − 0)²/25 + (y − 0)²/9 = 1
- Center (0, 0)
- Orientation horizontal (major axis along x)
- Vertices (-5, 0), (5, 0)
- Co-vertices (0, -3), (0, 3)
- Foci c = √(25 − 9) = 4 → (-4, 0), (4, 0)
- Eccentricity e = c/a = 0.8
- Area A = π·a·b = 47.1239
Result center (0, 0), foci (-4, 0), (4, 0), eccentricity 0.8
The focal distance comes from 25 − 9 = 16, giving c = 4 exactly. The foci therefore sit four units either side of the centre along the x-axis, comfortably inside the vertices at five units — which is always the case, since c is smaller than the semi-major axis.
The eccentricity is 4/5 = 0.8, a distinctly elongated ellipse. The area is 15π, larger than a circle of radius 3 and smaller than one of radius 5, which is the expected bracket.
The same ellipse turned on its side
Identical centre, but with a = 3 and b = 5. The two semi-axes have swapped, so the same shape now stands vertically.
Inputs Center x (h) = 0, Center y (k) = 0, Semi-axis a (x direction) = 3, Semi-axis b (y direction) = 5
- Equation (x − 0)²/9 + (y − 0)²/25 = 1
- Center (0, 0)
- Orientation vertical (major axis along y)
- Vertices (0, -5), (0, 5)
- Co-vertices (-3, 0), (3, 0)
- Foci c = √(25 − 9) = 4 → (0, -4), (0, 4)
- Eccentricity e = c/a = 0.8
- Area A = π·a·b = 47.1239
Result center (0, 0), foci (0, -4), (0, 4), eccentricity 0.8
The orientation line now reads vertical, and the vertices move to the y-axis while the co-vertices take the x-axis. The foci follow the major axis, so they too are now vertical — the single most useful consequence of reading that line first.
Eccentricity and area are unchanged at 0.8 and 15π. Both depend only on the two lengths and not on which direction each belongs to, which is a good check that the orientation logic has not leaked into quantities that should be independent of it.
Reading the result
What eccentricity is telling you
It measures departure from circularity on a scale from zero to one. Below about 0.3 an ellipse looks almost circular; above 0.8 it is visibly stretched. Being a ratio, it carries no units, so ellipses of very different sizes can share a value.
Vertices versus co-vertices
Vertices are the two points furthest from the centre and lie on the major axis; co-vertices are the two nearest and lie on the minor axis. Both pairs are on the curve, and it is only the major-axis pair that carries the name vertex.
When you would use this
Converting an equation into a description
Given a standard-form equation, the denominators are a² and b², so the semi-axes are their square roots. Feeding those in returns everything a question typically asks for.
Orbital and design geometry
Planetary orbits are ellipses with the primary body at one focus, and eccentricity is the standard way their shape is quoted. Elliptical arches and tanks use the same parameters, with the area formula giving the cross-section directly.
Assumptions and limitations
What this calculator assumes
- The ellipse is axis-aligned: the major axis runs parallel to x or to y, never at an angle.
- Both semi-axes are strictly positive; zero or negative values are rejected rather than treated as degenerate cases.
- The inputs are semi-axes — half-widths from the centre — not the full axis lengths.
Where it stops being the right tool
- Rotated ellipses are out of scope, so an equation with an xy cross-term cannot be entered.
- Input is in standard form only: a general conic equation must be completed to that form first.
- The perimeter is not reported. Unlike the area it has no elementary closed form and requires an elliptic integral or an approximation.
Common mistakes
Assuming a is always the semi-major axis
Why it happens. Textbooks usually write the standard form with a as the larger, so the letter comes to mean major. Here a names the horizontal direction, and b can easily be the longer of the two.
How to avoid it. Read the orientation line. It tells you which of the two the tool has taken as major, and every following figure is built on that.
Entering full axis lengths instead of semi-axes
Why it happens. A drawing usually shows the whole width and height, so those are the numbers at hand. Entering them doubles the ellipse and quadruples its area.
How to avoid it. Halve each measurement before entering it. The vertices line is the quickest check: it should sit at the distance you expect from the centre, not twice it.
Placing the foci on the minor axis
Why it happens. Both axes look equally plausible as homes for the foci, and the formula for c does not obviously say which.
How to avoid it. The foci always lie on the major axis, between the centre and the vertices. If a computed focus falls outside the curve, the axis assignment was reversed.
Frequently asked questions
What does eccentricity mean?
It is c/a, the focal distance divided by the semi-major axis, and it measures how far the ellipse departs from a circle. Zero is a perfect circle; values approaching one describe an increasingly flattened curve.
Which axis is the major one?
Whichever semi-axis is longer. If a exceeds b the major axis is horizontal; if b exceeds a it is vertical. The calculator detects this and states it before reporting anything that depends on it.
How is the area found?
A = π·a·b, the product of the two semi-axes and π. It does not depend on which is major, and it reduces to πr² when the two are equal — the circle being the case where both semi-axes are the radius.
Why is the perimeter not shown?
Because an ellipse has no elementary formula for its perimeter. Computing it requires an elliptic integral or one of several approximations, none of which is exact, so it is left out rather than reported with hidden error.