Ellipse Properties Calculator – Foci, Area & Eccentricity
An ellipse is the set of points whose distances to two fixed points (the foci) add to a constant.
In this calculator we use axis-aligned ellipses (no rotation / no \(xy\) term), so the major and minor axes are parallel to the coordinate axes.
1) Standard (center) forms
Let the center be \((h,k)\). If the major axis is horizontal (along x), then
\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\qquad a \ge b > 0.
\]
If the major axis is vertical (along y), the denominators swap:
\[
\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1,\qquad a \ge b > 0.
\]
2) Core properties
Focus distance and eccentricity. Define
\[
\begin{aligned}
c &= \sqrt{a^2 - b^2},\\
e &= \frac{c}{a}.
\end{aligned}
\]
The eccentricity satisfies \(0 \le e < 1\). If \(a=b\), then \(c=0\) and the ellipse is a circle.
Foci and vertices.
-
Horizontal major axis:
\[
\begin{aligned}
F_{1,2} &= (h \mp c,\;k),\\
V_{1,2} &= (h \mp a,\;k),\\
\text{Co-vertices} &= (h,\;k \mp b).
\end{aligned}
\]
-
Vertical major axis:
\[
\begin{aligned}
F_{1,2} &= (h,\;k \mp c),\\
V_{1,2} &= (h,\;k \mp a),\\
\text{Co-vertices} &= (h \mp b,\;k).
\end{aligned}
\]
Area.
\[
\text{Area} = \pi a b.
\]
Circumference (approximation). There is no simple exact elementary formula, but a common accurate approximation (Ramanujan) is
\[
\begin{aligned}
C &\approx \pi\Big(3(a+b)-\sqrt{(3a+b)(a+3b)}\Big).
\end{aligned}
\]
3) From general form to standard form (no rotation)
The calculator also accepts a general quadratic with no \(xy\) term:
\[
A x^2 + C y^2 + D x + E y + F = 0.
\]
Step 1. Complete the square in x and y to find the center.
\[
\begin{aligned}
A x^2 + D x &= A\left[\left(x+\frac{D}{2A}\right)^2 - \left(\frac{D}{2A}\right)^2\right],\\
C y^2 + E y &= C\left[\left(y+\frac{E}{2C}\right)^2 - \left(\frac{E}{2C}\right)^2\right].
\end{aligned}
\]
Step 2. Read off the center and the constant \(K\).
\[
\begin{aligned}
h &= -\frac{D}{2A},\qquad
k = -\frac{E}{2C},\\
A(x-h)^2 + C(y-k)^2 &= K,
\end{aligned}
\]
where \(K = A h^2 + C k^2 - F\). For a real ellipse (after normalizing signs), you must have \(A\) and \(C\) with the same sign and \(K>0\).
Step 3. Divide by \(K\) to get standard form.
\[
\begin{aligned}
\frac{(x-h)^2}{K/A} + \frac{(y-k)^2}{K/C} &= 1,\\
a_x &= \sqrt{\frac{K}{A}},\qquad b_y = \sqrt{\frac{K}{C}}.
\end{aligned}
\]
Then \(a=\max(a_x,b_y)\) and \(b=\min(a_x,b_y)\), and all properties follow from Section 2.
Example
Given \(\left(\frac{x}{4}\right)^2+\left(\frac{y}{3}\right)^2=1\), so \(h=k=0\), \(a=4\), \(b=3\).
\[
\begin{aligned}
c &= \sqrt{a^2-b^2}=\sqrt{16-9}=\sqrt{7},\\
e &= \frac{\sqrt{7}}{4},\\
\text{Area} &= \pi a b = 12\pi.
\end{aligned}
\]
