Ellipse Calculator

Analyze ellipse equations, dimensions, foci, directrices, area, perimeter, and graph.

Results are calculated automatically as you enter data.

Input method

Center h Center k Semi-major axis a Semi-minor axis b Major axis

Conversion and properties

Equation Enter valid ellipse data.

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Assumptions and limitations

Results are educational estimates, not professional engineering, construction, safety, scientific, legal, financial, or academic advice. No guarantee is made that outputs are accurate, complete, suitable, or error-free. Verify units, formulas, assumptions, tolerances, rounding, and limitations independently, and use qualified professional review for technical or safety-sensitive work. Use of this calculator is subject to our Terms of Use.

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What Is an Axis-Aligned Ellipse?

An ellipse is a smooth closed curve built around two fixed points called foci. For any point on the ellipse, the sum of the distances to the two foci stays constant. A circle is a special case of an ellipse where the two foci meet at the center.

An axis-aligned ellipse is an ellipse whose major and minor axes run horizontally and vertically on the coordinate plane. In other words, it is not tilted. This makes its equation easier to read because there is no $xy$ term and the key features can be found directly from the center, semi-axis lengths, and orientation.

The two most important lengths are the semi-major axis and the semi-minor axis. The semi-major axis, usually written as $a$, is half of the longest width of the ellipse. The semi-minor axis, usually written as $b$, is half of the shortest width. By convention, $a \ge b > 0$.

Why Axis-Aligned Ellipse Geometry Matters

Ellipse geometry connects the equation of a curve to the shape you see on a graph. Once an ellipse is in standard form, you can identify its center, vertices, co-vertices, foci, eccentricity, area, directrices, and approximate perimeter without relying only on a drawing.

This is especially useful when:

graphing conic sections in algebra or precalculus;
converting a quadratic equation into a readable standard equation;
checking whether a set of dimensions describes a real ellipse;
comparing how circle-like or elongated an ellipse is;
finding geometric features from a small set of inputs, such as a center and two semi-axis lengths.

The standard form also helps prevent a common mistake: confusing the full axis lengths with the semi-axis lengths. If the major axis length is $10$, then the semi-major axis is $a=5$, not $a=10$.

Key Terms to Know

Center: The point $(h,k)$ around which the ellipse is symmetric.
Major axis: The longer axis of the ellipse. Its full length is $2a$.
Minor axis: The shorter axis of the ellipse. Its full length is $2b$.
Semi-major axis: The distance $a$ from the center to a vertex along the major axis.
Semi-minor axis: The distance $b$ from the center to a co-vertex along the minor axis.
Vertices: The two endpoints of the major axis.
Co-vertices: The two endpoints of the minor axis.
Foci: Two points inside the ellipse, each $c$ units from the center along the major axis.
Focal distance: The distance $c$ from the center to either focus.
Eccentricity: The unitless value $e=\frac{c}{a}$ that describes how elongated the ellipse is.
Directrices: Lines associated with the focus-directrix definition of a noncircular ellipse.
Standard form: The equation form that shows the center and squared semi-axis lengths clearly.
General form: A quadratic equation form such as $Ax^2+Cy^2+Dx+Ey+F=0$ for the axis-aligned case supported here.

How Axis-Aligned Ellipse Geometry Works

An axis-aligned ellipse can be described in several equivalent ways. The goal is usually to convert the information you have into the same core set of properties: center, semi-major axis, semi-minor axis, orientation, focal distance, and standard equation.

Standard form for a horizontal ellipse

For an ellipse centered at $(h,k)$ with a horizontal major axis, the standard equation is:

$$ \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 $$

where $a \ge b > 0$.

The main features are:

center: $(h,k)$;
vertices: $(h-a,k)$ and $(h+a,k)$;
co-vertices: $(h,k-b)$ and $(h,k+b)$;
foci: $(h-c,k)$ and $(h+c,k)$.

Standard form for a vertical ellipse

For an ellipse centered at $(h,k)$ with a vertical major axis, the standard equation is:

$$ \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 $$

The main features are:

center: $(h,k)$;
vertices: $(h,k-a)$ and $(h,k+a)$;
co-vertices: $(h-b,k)$ and $(h+b,k)$;
foci: $(h,k-c)$ and $(h,k+c)$.

The larger denominator always belongs to the major-axis direction. If the larger denominator is under the $x$-term, the ellipse is horizontal. If the larger denominator is under the $y$-term, the ellipse is vertical.

Focal distance and eccentricity

The focal distance $c$ is found from the semi-axis lengths:

$$ c^2=a^2-b^2 $$

so:

$$ c=\sqrt{a^2-b^2} $$

Eccentricity is:

$$ e=\frac{c}{a} $$

For an ellipse, $0 \le e < 1$. When $e=0$, the shape is a circle. Values closer to $1$ describe a more elongated ellipse, although $e$ never reaches $1$ for a nondegenerate ellipse.

For a noncircular horizontal ellipse, the directrices are:

$$ x=h\pm\frac{a}{e} $$

For a noncircular vertical ellipse, the directrices are:

$$ y=k\pm\frac{a}{e} $$

When $e=0$, there is no finite directrix value because $\frac{a}{e}$ is undefined.

Area and perimeter

The area of an ellipse is:

$$ \text{Area}=\pi ab $$

The perimeter of an ellipse is more complicated than the perimeter of a circle. There is no simple elementary formula like $2\pi r$ that gives the exact ellipse perimeter in ordinary algebraic terms. This calculator reports an approximation:

$$ P\approx \pi\left(3(a+b)-\sqrt{(3a+b)(a+3b)}\right) $$

For stable floating-point calculation, the calculator evaluates the algebraically equivalent scaled form with $r=b/a$:

$$ P\approx \pi a\left(3(1+r)-\sqrt{(3+r)(1+3r)}\right),\qquad 0<r\le1 $$

This approximation is usually good for ordinary graphing and educational use, but it should not be treated as an exact symbolic perimeter.

Converting from general form

For an axis-aligned ellipse in the supported general form,

$$ Ax^2+Cy^2+Dx+Ey+F=0 $$

there is no $xy$ term. The coefficients $A$ and $C$ must be nonzero and must have the same sign. To convert to standard form, complete the square in $x$ and $y$.

First find the center:

$$ h=-\frac{D}{2A} $$

$$ k=-\frac{E}{2C} $$

Then compute the completed-square right-hand side:

$$ R=-F+Ah^2+Ck^2 $$

The equation becomes:

$$ A(x-h)^2+C(y-k)^2=R $$

After dividing by $R$, the standard-form denominators are:

$$ \frac{(x-h)^2}{R/A}+\frac{(y-k)^2}{R/C}=1 $$

Both denominators must be positive for a real ellipse. The larger denominator is $a^2$, and the smaller denominator is $b^2$.

Using a vertex and focus from the origin

A simplified vertex/focus setup can also describe an origin-centered ellipse. In this approach, the vertex coordinate gives the distance $a$ from the origin, and the focus coordinate gives the distance $c$ from the origin:

$$ a=\sqrt{v_x^2+v_y^2} $$

$$ c=\sqrt{f_x^2+f_y^2} $$

The focus distance must be smaller than the vertex distance:

$$ c

Then the semi-minor axis is:

$$ b=\sqrt{a^2-c^2} $$

This method is useful only for the supported origin-centered setup. The entered vertex and focus must each lie on a coordinate axis, and they must lie on the same axis; either may be on the negative side. It should not be used as a general method for shifted ellipses or tilted ellipses.

Examples of Ellipse Geometry in Practice

Example 1: Center and semi-axes

Suppose an ellipse has center $(2,-1)$, semi-major axis $a=5$, semi-minor axis $b=3$, and a horizontal major axis.

The standard equation is:

$$ \frac{(x-2)^2}{25}+\frac{(y+1)^2}{9}=1 $$

The focal distance is:

$$ c=\sqrt{5^2-3^2}=\sqrt{25-9}=4 $$

The eccentricity is:

$$ e=\frac{4}{5}=0.8 $$

The vertices are:

$$ (2-5,-1)=(-3,-1) $$

$$ (2+5,-1)=(7,-1) $$

The foci are:

$$ (2-4,-1)=(-2,-1) $$

$$ (2+4,-1)=(6,-1) $$

The area is:

$$ \pi ab=\pi(5)(3)=15\pi\approx 47.1239 $$

The perimeter approximation is:

$$ P\approx \pi\left(3(5+3)-\sqrt{(3\cdot5+3)(5+3\cdot3)}\right) $$

$$ P\approx 25.527 $$

Example 2: Completing the square from general form

Consider the equation:

$$ 9x^2+4y^2-54x+16y+61=0 $$

Here $A=9$, $C=4$, $D=-54$, $E=16$, and $F=61$.

Find the center:

$$ h=-\frac{-54}{2\cdot9}=3 $$

$$ k=-\frac{16}{2\cdot4}=-2 $$

Find $R$:

$$ R=-61+9(3^2)+4((-2)^2) $$

$$ R=-61+81+16=36 $$

Now divide by the denominators:

$$ \frac{(x-3)^2}{36/9}+\frac{(y+2)^2}{36/4}=1 $$

So the standard form is:

$$ \frac{(x-3)^2}{4}+\frac{(y+2)^2}{9}=1 $$

The larger denominator is under the $y$-term, so the ellipse is vertical. That gives $a=3$, $b=2$, and:

$$ c=\sqrt{3^2-2^2}=\sqrt{5} $$

The vertices are $(3,-5)$ and $(3,1)$, and the foci are $(3,-2-\sqrt{5})$ and $(3,-2+\sqrt{5})$.

Example 3: Vertex and focus distances from the origin

Suppose the vertex coordinate is $(0,4)$ and the focus coordinate is $(0,3)$. The center is treated as the origin.

The vertex distance is:

$$ a=\sqrt{0^2+4^2}=4 $$

The focus distance is:

$$ c=\sqrt{0^2+3^2}=3 $$

Since $c<a$, this can describe an ellipse. The semi-minor axis is:

$$ b=\sqrt{4^2-3^2}=\sqrt{7} $$

Because the vertex is vertical, the standard equation is:

$$ \frac{x^2}{7}+\frac{y^2}{16}=1 $$

The eccentricity is:

$$ e=\frac{3}{4}=0.75 $$

The directrices are:

$$ y=0\pm\frac{4}{0.75} $$

$$ y=\pm\frac{16}{3} $$

How to Interpret the Result

The standard equation is usually the most important result. It tells you the center and shows which axis is major by placing the larger denominator under the major-axis variable.

The center shows the point of symmetry. If the equation contains $(x-2)^2$, the $x$-coordinate of the center is $2$. If it contains $(x+2)^2$, the $x$-coordinate of the center is $-2$ because $(x+2)^2=(x-(-2))^2$.

The semi-major and semi-minor values are half-lengths. The full major axis is $2a$, and the full minor axis is $2b$.

Eccentricity is unitless. A value near $0$ means the ellipse is close to circular. A larger value below $1$ means the foci are farther from the center and the ellipse is more elongated.

Area is reported in square coordinate units because it comes from $\pi ab$. If your coordinate unit is meters, the area is in square meters. If your input values are unitless, the area is also interpreted in squared arbitrary units.

The perimeter value is approximate. It is useful as a practical estimate of the distance around the ellipse, but it is not the same kind of exact result as the standard equation or the area formula.

Vertices are the endpoints of the major axis. Co-vertices are the endpoints of the minor axis. Foci are inside the ellipse on the major axis, $c$ units from the center. Directrices are vertical for horizontal ellipses and horizontal for vertical ellipses.

Common Mistakes and Misconceptions

Entering full axis lengths instead of semi-axis lengths. If the full major axis is $12$, enter $a=6$ when a semi-major axis is requested.
Assuming $a$ and $b$ can be in any order. In standard ellipse notation, $a$ is the semi-major axis and $b$ is the semi-minor axis, so $a\ge b$.
Forgetting that orientation matters. A horizontal ellipse and a vertical ellipse can have the same $a$ and $b$ values but different equations.
Reading signs in the center incorrectly. The term $(x+4)^2$ means $h=-4$, not $h=4$.
Using a rotated ellipse. Equations with an $xy$ term describe a rotated conic and are outside the supported axis-aligned setup.
Not moving all terms to one side in general form. The supported form is $Ax^2+Cy^2+Dx+Ey+F=0$.
Using foci mode for a shifted ellipse. The simplified vertex/focus method assumes the center is the origin.
Entering a focus distance greater than or equal to the vertex distance. For an ellipse, $c<a$ must hold.
Treating the perimeter estimate as exact. Ellipse perimeter calculations generally rely on approximations or special functions.
Mixing units. Coordinates, semi-axis lengths, vertices, and foci should use the same unit scale.

When to Use Axis-Aligned Ellipse Geometry

Use axis-aligned ellipse geometry when you need to describe or analyze an ellipse whose axes are parallel to the coordinate axes.

Common uses include:

writing the equation of an ellipse from its center and dimensions;
graphing an ellipse from a standard equation;
finding vertices, co-vertices, and foci for a conic-section problem;
converting a supported quadratic equation into standard form;
checking whether a set of general-form coefficients represents a real ellipse;
comparing ellipses by eccentricity, area, or approximate perimeter.

If the ellipse is tilted, has an $xy$ term, or comes from a context that requires high-precision perimeter measurements, use a more specialized method.

Limitations and Things to Keep in Mind

This calculator is designed for axis-aligned ellipses. It does not handle rotated ellipses or equations that contain an $xy$ term.

In dimensions mode, the inputs $a$ and $b$ are semi-axis lengths, not full axis lengths. The values must be positive and must satisfy $a\ge b$. If $b>a$, the calculator shows an input error; it does not reorder the axes.

In general-form mode, the supported equation is:

$$ Ax^2+Cy^2+Dx+Ey+F=0 $$

The coefficients $A$ and $C$ must be nonzero and have the same sign. After completing the square, the denominators must be positive. Otherwise, the equation does not describe a real ellipse in the supported format.

In vertex/focus mode, the center is assumed to be $(0,0)$. The entered vertex and focus are treated by their distances from the origin. This means the method does not fully verify every geometric detail that would be needed for an arbitrary shifted or rotated ellipse.

No unit conversion is built into the calculation. All coordinate and length inputs should use the same implicit unit.

Displayed values may be rounded. Only values that are exactly zero display as $0$; very large or very small nonzero values may display in scientific notation. This formatting helps keep results readable, but it means a displayed decimal may not show every internal digit.

The calculator uses finite JavaScript numbers. It rejects dimensions when the squared axis lengths, area, or perimeter approximation cannot be represented reliably. For extremely unequal axes, rounding can make the displayed eccentricity appear as $1$, even though a nonzero semi-minor axis was entered.

The directrix formula involves $\frac{a}{e}$. When an ellipse is circular, $e=0$ and there is no finite directrix. When an ellipse is very close to circular, $e$ can be very small, so the directrix values can become very large.

How to Use This Calculator

Choose the calculator mode: dimensions, general form, or foci.
For dimensions mode, enter the center coordinates $h$ and $k$, the two positive semi-axis lengths, and the orientation.
For general-form mode, enter $A$, $C$, $D$, $E$, and $F$ for the equation $Ax^2+Cy^2+Dx+Ey+F=0$.
For foci mode, enter a vertex coordinate and a focus coordinate relative to the origin. Make sure the focus distance is smaller than the vertex distance.
Review the standard equation, center, semi-axis lengths, eccentricity, area, graph, important points, directrices, focal distance, and perimeter approximation.
Use the graph download option when you need to save the plotted ellipse image.

Frequently Asked Questions

What is the difference between the major axis and the semi-major axis?

The major axis is the full longest width of the ellipse. The semi-major axis is half of that width, measured from the center to a vertex. If the major axis length is $2a$, the semi-major axis length is $a$.

How do I know whether an ellipse is horizontal or vertical?

Look at the larger denominator in standard form. If the larger denominator is under the $x$-term, the major axis is horizontal. If the larger denominator is under the $y$-term, the major axis is vertical.

Can an ellipse have equal semi-axis lengths?

Yes. When $a=b$, the ellipse is a circle. In that case, the focal distance is $c=0$, eccentricity is $e=0$, and there is no finite directrix.

Why is the perimeter only approximate?

Unlike the circumference of a circle, the perimeter of an ellipse does not simplify to a basic elementary formula. Practical calculations often use approximations or special functions. The displayed perimeter is therefore an estimate, not an exact symbolic value.

Can I use an equation with an $xy$ term?

No. An equation with an $xy$ term generally represents a rotated conic. This calculator focuses on axis-aligned ellipses, so the supported general form includes $x^2$, $y^2$, $x$, $y$, and a constant term, but not an $xy$ term.

Why does $(x+3)^2$ mean the center has $h=-3$?

Standard form uses $(x-h)^2$. If the expression is $(x+3)^2$, it can be rewritten as $(x-(-3))^2$. That means the center’s $x$-coordinate is $-3$.

Sources and References

Books and Textbooks

Jay Abramson. College Algebra 2e. OpenStax, 2021. Chapter 8, Section 8.1, “The Ellipse”; Chapter 8 Key Concepts. Section 8.1 and Chapter 8 Key Concepts.
Gilbert Strang and Edwin “Jed” Herman. Calculus Volume 2. OpenStax, 2016. Chapter 7, Section 7.5, “Conic Sections.” Section 7.5.

Online and Official Sources

Eric W. Weisstein. “Ellipse.” MathWorld—A Wolfram Web Resource, accessed June 28, 2026. https://mathworld.wolfram.com/Ellipse.html.
National Institute of Standards and Technology. “§19.30 Lengths of Plane Curves.” NIST Digital Library of Mathematical Functions, accessed June 28, 2026. https://dlmf.nist.gov/19.30.