The radius vector from the Sun to a planet sweeps out equal areas in equal periods of time.
(Based on a diagram by John P. Oliver, a fellow graduate student and astronomer; original no longer online)
Above, each area is equal but the arcs that sweep out the areas are not; closer to the Sun the arcs are longer, because the planet moves faster (see below); further from the Sun the arcs are shorter, because the planet moves slower.
Close to the Sun a given arc sweeps out a small area; further away it sweeps out a larger area. This means that it takes less time to cover an arc close to the Sun, so the planet must be moving faster, and more time to cover an arc further from the Sun, so the planet must be moving slower. In other words, the Law of Areas implies that as the planets move away from the Sun they slow down, and as they move toward it they speed up. For small changes in distance (loweccentricity orbits), the change in speed is about the same (in percentage terms) as the change in distance; in other words, for each percent that a planet moves closer to the Sun it speeds up by about one percent, and for each percent that it moves away from the Sun it slows down by about one percent. As an example, as the Earth moves 1.7% closer to the Sun than on average in January, it speeds up by 1.7% compared to its average speed; and as it moves 1.7% further from the Sun than on average in July, it slows down by 1.7% compared to its normal speed (see Cassini Measures the Motion of the Sun for a diagram). For larger changes in distance things are not so simple, except at perihelion and aphelion, where the motion is perpendicular to the radius vector; at those points the speed is inversely proportional to the distance; so if an object were 10 times further from the Sun at aphelion than at perihelion, it would be going 10 times slower. For some comets aphelion is as much as 100,000 times further out than perihelion, and although the comet moves nearly 200 miles per second at perihelion, at aphelion, going 100,000 times slower, it only moves about 10 feet per second and takes 300 years to move as far as it would in one day at perihelion.
It is possible to calculate the speed at any point in the orbit using the principle of conservation of energy, in which the kinetic energy of the planet (calculated from its speed and mass) plus its potential energy (calculated from its mass and distance from the Sun) always have a sum equal to a constant value. Solving the resulting equation yields a formula (called the Vis Viva Equation) relating the speed to the orbital size and eccentricity and the current distance from the Sun. (The mathematical and physical concepts involved are beyond the introductory level of most high school or lower division astronomy courses, so a discussion that is both clear and accurate requires some work, and as a result will be added to this or a supplementary page at some future date).
As a planet moves from aphelion to perihelion it sweeps out half the area of the orbit, and the same area is swept out as it moves from perihelion to aphelion, so the times required to go from one point to the other are equal. This is because on the way in the planet is speeding up and on the way out it is slowing down, and for every point on the inward journey where the planet has a particular speed there is a symmetrical point on the outward journey where the planet has the same speed; so it covers the equal arcs at different speeds at different times, but with the same average speed over each halforbit shown above.
In the diagram above the planet moves from one end of the minor axis to the other end, sweeping out a small area as it moves through perihelion. Covering the same distance (half the perimeter of the ellipse) as it moves through aphelion it sweeps out a much larger area. This is because at every moment the planet spends on the perihelion halfellipse it is closer to the Sun, moving faster; and at every moment it spends on the aphelion halfellipse it is further from the Sun, moving slower. So it takes much less time to cover the perihelion half of the orbit than it does to cover the aphelion half. As a result, objects with very eccentric orbits (such as comets) spend far more time far from the Sun than they do, near it (remember the example above, where it took a comet 300 years to cover the same distance at aphelion as it covers in one day at perihelion; obviously, such a comet would spend a far longer time far from the Sun, where it is moving slower, than near the Sun, where it is moving much faster).
The Force Behind The Law: A Cautionary Note
The reason that the planets move this way is that the force that keeps them in their orbits points at the Sun. When they are moving away from the Sun its pull is backwards and sideways relative to their direction of motion, so they slow down and curve toward the Sun. When they are moving toward the Sun its pull is forward and sideways relative to their direction of motion, so they speed up and curve toward the Sun. Neither the nature of the force, its strength, or the way its strength changes with distance alters that statement. As a result, the Law of Areas would be true no matter what kind of force the Sun exerted on the planets, as long as that force pointed directly toward the Sun.
The force that the Sun exerts on the planets is the same as the gravitational force that the Earth exerts on us; and as discovered by Isaac Newton, it is an inversesquare law, meaning that it gets weaker as the planet moves away from the Sun, and stronger as the planet moves toward the Sun. Students often mistakenly presume or state that planets slow down as they move away from the Sun because the Sun's force is getting weaker (with the implication that if they didn't go slower its force couldn't hold onto them), and that planets speed up as they move toward the Sun because the Sun's force is getting stronger, and a stronger force makes them go faster than a weaker one would. This sounds reasonable, but is a totally incorrect analysis of the Law of Areas. No matter how the Sun's force changes with distance  getting weaker as the inverse square of the distance or in any other way, not changing, or even getting stronger (as in the case of springs, which pull harder the more they are stretched)  only the shape of the orbit would be affected. The Law of Areas would be completely unaffected, because all that it depends on, as stated above, is that the force is backward (slowing the planet down) as it moves outward, and is forward (speeding the planet up) as it moves inward.
