It's easy to think of an ellipse as "a squashed circle," but an ellipse is actually defined in a slightly different way than a circle. Stick two thumbtacks into a board (call these two points the foci), and loop a string loosely between them. Pull the string taut with a pencil and trace all the way around, and the curve that pencil draws is an ellipse.
Pick any point on the curve drawn this way, and adding up its distances to the two foci (F₁, F₂) always gives the same total — exactly the length of the string. That's the true definition of an ellipse. If the two foci were to completely merge into the same spot (both thumbtacks in one place), this definition would become exactly the definition of a circle ("the set of points at the same distance from a center"). In other words, a circle is a special ellipse whose two foci have merged.
The number that describes how far an ellipse strays from being a circle is called eccentricity (e). It's the ratio of the semi-major axis (the direction of the ellipse's longest radius, a) to half the distance between the foci (c), calculated as e = c ÷ a. When eccentricity is 0, the two foci completely merge into a circle, and as eccentricity gets closer to 1, the ellipse gets flatter and more elongated. The orbital eccentricities of most planets in our solar system are extremely close to 0 (nearly circular ellipses), close enough that it's hard to tell them apart from a circle by eye.
The area formula for an ellipse is deeply connected to the area formula for a circle too. The area of an ellipse is found as π × a × b (a is the semi-major axis, b is the semi-minor axis), and if a and b become equal (in other words, if the ellipse becomes a circle), this formula becomes exactly π × r × r = πr². The area formula for a circle turns out to be a special case of the ellipse area formula all along.
The focus property of an ellipse gets put to fascinating use in real life too. The German astronomer Kepler discovered that planets orbit the Sun not in a perfect circle but along an elliptical orbit, and in this orbit, the Sun sits not at the center of the ellipse but at one of its foci (Kepler's first law). That's why a planet moves faster when it's close to the Sun and slower when it's far away. Another fun example is an elliptical building called a "whispering gallery." When sound reflects off a point on an elliptical wall, the angle at which sound from one focus hits the wall and the angle at which it heads toward the opposite focus are always exactly equal — so a soft whisper from one focus, after reflecting, comes through loud and clear at the opposite focus.
On our activity page, you can move point P as if actually pulling a string taut and check whether the sum of its distances to the two foci really stays constant, watch how flat the ellipse gets as you adjust eccentricity, change the semi-major and semi-minor axes separately to see how the area formula connects to the area formula for a circle, and try out planetary orbits and whispering galleries for yourself.