Map scale confusion.

[Beron]

Now, I'm not sure where to post this, so I felt it'd be safest to do so here.

I have a bit of a problem with map scaling...and math. To make a long story short, it was a biblical miracle I ever passed math at all. Numbers and me don't mix, and so when I come across a mathematical problem I panic. This is what it is.

My 'world' is around the size of Earth, but just a bit larger. The surface area is:

150,000,000 km^2 for Land
365,000,000 km^2 for Water

And so a total of 515,000,000 km^2 for the entirety of the planet.

My goal is to work out how many km = to 1cm on the map. Apparently the proper way to do this is to work with random numbers, like 1:10000000, but I need the scale to be the exact size of earth, it's vital to the lore behind the map.

To make life easier (just a tad) I've separated this "pangea" into different continents, still, I don't think the size of the number will help at all.

The Map is being worked on an A4 sketch pad, portrait, and so is 624cm^2 in area.

If any of you are knowledgeable in this field of work, I'll thank you a thousand times over (using a gif of course xD.)

I apologise if I come across as arrogant, but this thing has been eating my coconut for days.

Thanks alot, Beron.

[Hai-Etlik]

Most global maps don't preserve areas anyway, and the only projection that is equal area and fits exactly into a rectangle is Cylindrical Equal Area.

"Apparently the proper way to do this is to work with random numbers, like 1:10000000" Uh, randomness does not enter into it, and a number by itself is not random, it's the process that produced it that is, not that 1/10000000 is a number most people would think of as being "random" anyway or that randomness would be in any way relevant to scaling maps.

"but I need the scale to be the exact size of earth" This makes absolutely no sense. I have no clue what you are trying to accomplish. If you hadn't already said the planet is larger than Earth, I'd assume you meant they need to be the same size but as it is, I have no clue.

I'm sorry but if you have a clear idea of what you want, you aren't expressing it effectively.

[Azélor]

515 000 000 km2 makes for a map of something like 16 047 cm by 32 094 cm but that is huge. At a resolution of 300 you would be able to have a document of 160 by 320 cm. That's 18898 x 37795 pixels and by expirience, depending on your computer, is the limit concerning the size. Even with that, you might encounter performance issues if you have a lot of stuff on your map. So that's probably the maximum you can go.

[feanaaro]

Perhaps the original request is a little confuse, but I think I get what Beron wants. Now, the area of a sphere (no planet is a perfect sphere, but close enough) is 4πR^2. This means that if the total area of the planet is 515 000 000 km2, then its radius is the square root of the Area/4π, which is approximately equal to 6402 km (I guess it's ok to be approximate, since the planet is not a perfect sphere to begin with), which is in fact just a little bigger than Earth's. A radius of 6402 gives a circumference (2πR) of approximately 40225 km. So, let's say that the Equator of this planet is 40225 km long. Then, assuming you are using a rectangular projection in which the scale is constant at the equator (which is most of them, I guess, or at least most of those that are most commonly used), that is the width of the map at the equator. If you use an A4 size in landscape mode, that means that 297 mm = 40225 km, thus 1mm =~ 135,4 km, 1cm =~ 1354 km. 1cm^2 =~ 1833316 km^2. If you count by pixels instead of mm just make the appropriate proportion with the width of 40225km.
Unless I made some errors with the calculation, but you get the idea anyway.

[Azélor]

Oups, my numbers considered that the Earth was a rectangle. Your are pobably right Feanaaro and I think it's easier to use pixels instead of cetimeters. It's irrelevant if you don't plan to print it.

[Beron]

This makes absolutely no sense. I have no clue what you are trying to accomplish. If you hadn't already said the planet is larger than Earth, I'd assume you meant they need to be the same size but as it is, I have no clue.

Intoxication and silly 5am ideas make for incoherence, sorry :-) I'll try better with good sleep and a slight hangover xD

So, thank you ever so much for the replies, guys/gals and I apologise for my complete incompetence with maths. Seriously, that subject is like carbs to an endomorph for me.

So, if I'm not mistaken. If I consider the entirety of the map, including sea + land, 1cm will approximately equal to 1354km?

It seems to me that feanaaro feels that I'm working with a sphere? I know I said 'planet' but if it makes things easier, the map I'm drawing works like a typical 'map' of the globe. Unless that's completely irrelevant.

If that doesn't make things easier, maybe splitting the globe into sections and working at them individually will help? For example:

The world at its current age is akin to Earth's pangea: it's one super continent. I considered the future and made outlines to what will eventually become 8 separate land masses. The largest being 30million kilometres square (around the size of Africa) and the smallest 10million kilometres square.

Again, I apologise for the inconvenience, but I still haven't grasped it fully yet. Since 1cm = 1345km on a global scale, how can I equate 30million kilometres square on a continental map?

[feanaaro]

If your equator is 40225 km long, your planet is roughly spherical, and you are working on an A4 sheet, then 1cm =~ 1354 km. This is really simple and there is no need of making the planet a flat square or splitting it in parts or anything.
1cm^2 = 1 833 316 km^2, therefore 30 million km^2 =~ 16.4cm^2, or a little more than a 4cm*4cm square.
However, with all due respect, if given the km^2 size of 1cm^2 on the map you are not able to calculate how many cm^2 you would need for a given number of km^2, perhaps you should not worry about these things in the first place, and just mapping without bothering to consider scale (which is how humans have mapped the world for most of their history, anyway).

[lastofthemany]

There is an issue with the scale calculation above. If you are talking about a rectangular map of a spherical world, then the scale of 1cm = 1354 km is only true at the equator. The closer to the poles you get, the greater the distortion. In effect your ratio scale approaches 1to1 till 1cm = 1cm at the poles. That is because the top and bottom neat lines of the map are actually two different points, the north and South Pole respectively. To make a long story short, follow Feanaaro's advice. Scale is elusive on a global map that isn't an actual globe, unless you have a program that allows you to recalculate your map at different projections.