I worked this out using the Jump Shot skill, which leaps the player exactly 17.78 metres at max range. We are told by Anet that one unit of range is equal to 1 inch, so since Jump Shot has a range of 700 units you would travel 1778 centimetres.
What I did was go to the Mustering Ground Waypoint in the Black Citadel and make 15 Jump Shots east, before taking a screenshot of the map and measuring how big that distance (266.7 metres in-game) was on my screen: it ended up being 5.82 cm according to PowerPoint. After that, it was just a case of drawing a lot of lines and measuring out each zone.
Here are the areas of each of the 26 PvE zones according to my calculation:
Regrettably I didn’t have enough time to get any accurate measurements for WvW, but I did do a very rough estimate which suggested that WvW’s total area is about 28 km2.
Now I'm pretty sure later on someone will measure things more accurately, but I hope this is interesting enough for now.
Edit: I'd also estimate the total planetary surface area (from the big globe map) to be 3998 square km, since the circumference appears to be 103 km at the equator.