Could someone help me calculate the areas in this map?

Question

Map of my book - Please click here to see it

So, I'm writing a book and I'm trying to be really detailed. Right now I'm writing a wiki of it and I want to specify the areas of continents and cities.

But I'm having a lot of trouble doing this. I'm not good at math and the shapes are complex, so I won't find an equation to solve everything.

Is there a way to solve this? More than just the answers, I really need to understand so I can calculate this alone later.

If it helps, each one of the rectangles in the grid have 3 million km² (I built it in Illustrator with 1cm height and 2cm width).

Answer 1

The way the map looks (equidistant latitudes), you should probably weigh pixels at latitude $\phi$ with a factor of $\cos\phi$. That is (as your image is $539\times 508$ pixels):

area = 0.0
planetsurface = 0.0
for y=0 to 507
   weight = sin(pi*(y+0.5)/508)
   planetsurface = planetsurface + weight*539
   for x=0 to 538
      if PixelAt(x,y) is land then
         area = area + weight
      endif
   next
next
print area*100/planetsurface, " percent of the surface are land"