Ellipse Eccentricity Calculator – e = c/a (Free Tool)
An ellipse is the set of points whose distances to two fixed points (the foci)
have a constant sum. This tool computes the ellipse’s eccentricity \(e\),
a number in the range \(0\le e<1\) that measures how “stretched” the ellipse is.
1) Standard axis-aligned ellipse
The most common (non-rotated) form is:
\[
\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1,
\qquad a\ge b>0,
\]
where \((h,k)\) is the center, \(a\) is the semi-major axis length, and \(b\) is the
semi-minor axis length.
If the larger denominator is under \((x-h)^2\), the major axis is horizontal.
If the larger denominator is under \((y-k)^2\), the major axis is vertical.
2) Eccentricity \(e\) and focal distance \(c\)
The focal distance \(c\) (center-to-focus distance) satisfies:
\[
c^2=a^2-b^2 \quad\Longrightarrow\quad c=\sqrt{a^2-b^2}.
\]
The eccentricity is:
\[
\boxed{e=\frac{c}{a}}
\qquad\text{and equivalently}\qquad
\boxed{e=\sqrt{1-\frac{b^2}{a^2}}}.
\]
- If \(a=b\), the ellipse is a circle and \(e=0\).
- As \(b\) becomes much smaller than \(a\), the ellipse becomes more stretched and \(e\) moves closer to \(1\).
3) Foci and directrices
Once \(c\) is known, the foci are easy to locate.
-
Horizontal major axis (major along \(x\)):
\[
F_1=(h-c,k),\qquad F_2=(h+c,k).
\]
-
Vertical major axis (major along \(y\)):
\[
F_1=(h,k-c),\qquad F_2=(h,k+c).
\]
An ellipse also has two directrices (used in the focus-directrix definition of conics).
They are a distance \(\frac{a}{e}\) from the center along the major axis:
-
Horizontal major axis:
\[
\boxed{x=h\pm\frac{a}{e}}.
\]
-
Vertical major axis:
\[
\boxed{y=k\pm\frac{a}{e}}.
\]
For a circle (\(e=0\)) the directrix description is not used, so the directrices are typically treated as “not applicable”.
4) From a standard equation (denominators)
If you are given:
\[
\frac{(x-h)^2}{A}+\frac{(y-k)^2}{B}=1,
\]
then the axis lengths are \(\sqrt{A}\) and \(\sqrt{B}\). The semi-major axis is the larger of these:
\[
a=\max(\sqrt{A},\sqrt{B}),\qquad b=\min(\sqrt{A},\sqrt{B}).
\]
Then compute \(c=\sqrt{a^2-b^2}\) and \(e=\frac{c}{a}\).
5) From a general non-rotated equation (extra mode)
If there is no \(xy\) term, the general form is:
\[
Ax^2+Cy^2+Dx+Ey+F=0.
\]
Completing the square gives:
\[
A(x-h)^2 + C(y-k)^2 = K,
\]
where
\[
h=-\frac{D}{2A},\qquad k=-\frac{E}{2C},\qquad
K=\frac{D^2}{4A}+\frac{E^2}{4C}-F.
\]
Dividing by \(K\) yields the standard form:
\[
\frac{(x-h)^2}{K/A}+\frac{(y-k)^2}{K/C}=1,
\]
so the denominators are \(A_x=K/A\) and \(A_y=K/C\), and you proceed as in the standard-equation method.
Example
Given semi-axes \(a=5\), \(b=3\):
\[
c=\sqrt{a^2-b^2}=\sqrt{25-9}=4,\qquad
e=\frac{c}{a}=\frac{4}{5}=0.8.
\]
The graph shows the ellipse in square units. Optional overlays include the center, foci, directrices, and the major/minor axes.
