An ellipse is the locus of a point which moves so that the sum of its distances (\(d_1 + d_2\) in the diagram) from two fixed points called the foci (F and G in the diagram is constant: \(d_1 + d_2 = \text{constant}\).

Ellipses look like squashed circles.

Put a piece of paper on top of a board. Stick drawing pins into the paper at two points. Make a loop out of a piece of string, and place it around the two drawing pins.

Take your trusty pencil, put it inside the loop, and use it to stretch out the loop of string so that it forms a triangle.

Keeping the string stretched, draw a curve. Your pencil will trace out an ellipse.

If you only use one drawing pin, you get a circle. A circle is an ellipse in which the two foci are in the same place.

The axes of an ellipse cut it in half. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest diameter of the ellipse. Each passes through the centre of the ellipse.

As their names suggest, the semi-major axis is half of the major axis of the ellipse, and the semi-minor axis is half of the minor axis of the ellipse.

The formula for the area of an ellipse is exactly what you would expect: \(\text{Area} = \pi a b\), where \(a\) and \(b\) are the lengths of the semi-minor and semi-major axes. The formula for the area of a circle is a special case of this as both semi-major and semi-minor axes of a circle have the same length (the radius of the circle): \(\pi r^2\).

When you cut a slice through a cone, parallel to the base, you get a circle. When you slice through a cone, at a slight angle to the base, you get an ellipse.

This is why we say that ellipses are conic sections. The article on conic sections will tell you more about the other conic sections.

Each conic section is assigned a number, called the eccentricity, that tells you how close it is to being a circle. A circle has eccentricity \(0\), and the eccentricity of an ellipse is somewhere between \(0\) and \(1\).