An ellipse is a cousin of the circle. From its two foci, eccentricity, and area formula to real-world examples — change the values with sliders and explore the properties of an ellipse.
An ellipse isn't just a squashed circle — it's a shape with its own precise definition. Loop a string loosely around two thumbtacks (the foci), pull it taut with a pencil, and trace all the way around: the resulting shape shows exactly the property that, for any point on an ellipse, the sum of its distances to the two foci is always the same.
As the two foci get closer and closer together and eventually merge into a single point, the ellipse naturally becomes a circle. In other words, a circle is "a special ellipse whose two foci have merged into one." Even Earth's orbit around the Sun isn't a perfect circle — it's an ellipse with a very small eccentricity.
Change the distance between the foci and the position of point P with the sliders, and see for yourself how much flatter the ellipse gets as eccentricity increases, and how the area (πab) changes.
Loop a string around two pins (foci F₁, F₂) and trace it out with a pencil pulled taut, and you get an ellipse. Move point P anywhere on the ellipse, and the sum of its distance to F₁ and its distance to F₂ is always the same — that's exactly the length of the string (2a).
Eccentricity e = c/a shows how far the ellipse strays from being a circle. Close to 0, it stays nearly circular; close to 1, it gets very flat.
Adjust the semi-major axis (a) and semi-minor axis (b) separately. If a and b become equal, the ellipse turns into a circle, and the area formula becomes exactly πr².
Planetary orbits and "whispering galleries" both rely on the focus property of an ellipse. See it for yourself below.