What is an Ellipse Calculator?
An ellipse is a closed curve where the sum of distances from two fixed points (the foci) to any point on the curve is constant. This calculator takes the two defining measurements of an ellipse — the semi-major axis a (half the longest diameter) and the semi-minor axis b (half the shortest diameter) — and instantly returns the area, perimeter, eccentricity and focal distance.
How to use it
Enter the semi-major axis a and the semi-minor axis b in any consistent unit (cm, m, inches, etc.). Press calculate. The area comes out in square units and the perimeter and focal distance in the same linear units you entered. If you enter the values in the wrong order, the eccentricity and focal distance are still computed correctly because the calculator automatically uses the larger value as the major axis.
The formulas explained
The area of an ellipse is exact: \(A = \pi\,a\,b\). When \(a = b\) the ellipse is a circle and this reduces to \(\pi r^{2}\). The perimeter has no simple closed form, so we use Ramanujan's famous approximation $$P \approx \pi\left[\,3(a+b)-\sqrt{(3a+b)(a+3b)}\,\right]$$ which is accurate to better than one part in ten million for typical shapes. The eccentricity \(e = \sqrt{1-\frac{b^{2}}{a^{2}}}\) describes how stretched the ellipse is: 0 means a perfect circle, while values approaching 1 describe an increasingly flattened shape. The focal distance \(c = \sqrt{a^{2}-b^{2}}\) gives the distance from the center to each focus.
Worked example
For \(a = 5\) and \(b = 3\): Area $$A = \pi\cdot 5\cdot 3 = 15\pi \approx 47.12 \text{ square units.}$$ Perimeter $$P \approx \pi\left[3(8) - \sqrt{18\cdot 14}\right] = \pi\left[24 - \sqrt{252}\right] \approx \pi\cdot 8.124 \approx 25.527 \text{ units.}$$ Eccentricity $$e = \sqrt{1 - \frac{9}{25}} = \sqrt{0.64} = 0.8.$$ Focal distance $$c = \sqrt{25 - 9} = \sqrt{16} = 4.$$
FAQ
What units does it use? Any unit you like — just keep \(a\) and \(b\) in the same unit; the area is returned in that unit squared.
Why is the perimeter approximate? The true ellipse perimeter requires an elliptic integral with no elementary closed form. Ramanujan's formula is an extremely accurate, fast approximation.
What does eccentricity of 0 mean? An eccentricity of 0 means \(a = b\), so the ellipse is actually a circle.
