I am unsure what the official rules are for measuring area of effects for the different types of areas. Here are my assumptions (please correct them) with a bit of visual help from this chart from pathfinder
Line: pretty straight forward (pun intended)
Cube: A bit like the area bursts in 4e, although the point of original has changed, pretty clear
,
Cone: ok now it gets interesting. For a 15' cone, I wondering if the proper area is: 1 square in front row, 3 squares in 2nd row and...3 squares in 2nd row? For a 30' cone the squares per row as we go down the cone would be: 2,4,6? That doesn't agree with the above chart but it sounds logical. Also can the point of origin be 'off-center' like in the second 30' cone blast shown above?
What is the practical difference between a cylinder area of effect and a sphere area of effect? Both seem to work the same for an identical level of elevation (sphere's volume would decay as we moved up or down in elevation as compared to a cylinder of course). Are their areas of effect like the ones posted above for the 10' radius and 20' radius?
I use 3E AoE model too (we have transparent spreadsheet) but the only thing we use differently is radius and lines filling a number of 5' space/squares due to having each square being 5', including diagonals. So a 10 feet radius around you fills every squares within two of you for exemple.
For effect originating from a point in space rather a creature, we use dots where two lines cross, so a 10 feet radius originating from a point in space fills a 2x2 square for exemple.
why does mesureing diagnal take squares away?
diagnal squares also measure 5'
i have always taken it as any square touched is a square hit, so if it goes 30' it would go 30' diagnal too, your example shows the PC losing 10' on cones, and 40' on burst area/aura.
EDIT: i suppose as long as it is equal to both PC and bad guy, it wouldn't matter in the long run.
measuring diagonally takes away squares because the hypotenuse of an isoceles right triangle of 1 unit in length on its other sides is 1.41 (square root of 2), which is almost a square and a half in length...
D&D gird rules I've seen omit this fact for movement in order to simplify. The problem encountered isn't so much for the line spells where they can easily be done one way or another, but for the diagonal cone spells like the first 30' cone spell in the image. If the spell was diagonally 6 squares long, it would contain many more squares (21) than the 2 sq, 4 sq, 6 sq per row for non diagonal 30' cone (the above image doesn't show that). That's why I'd be tempted to make the diagonal cone blast 5 squares on each of its sides.
It's the problem with taking reality and reducing it to a series of cubes. Which is really what you are doing with D&D.
Diagonalls at 22.5 degrees add an extra .59 feet every 5 foot square.
Soif you are doing a 22.5 degree angle, you need to go 9 squares before you gain 5 feet. (assuming round down)
However, at 45 degrees, you gain 2.07 feet every 5 feet, so you have to go 3 squares before you gain 5 feet. (assuming round down)
d20 made it so every other counts as 10 feet. Which is actually significantly less. If you use that rule you are always better off going at 22.5 degree angles to go anywhere (or move like a knight on a chessboard.)
Now at 22.5 you should have this
Squares - Distance (at 22.5 degrees) - REAL distance
1 - 5' - 5.59
2 - 10' - 11.18
3 - 15' - 16.77
4 - 20' - 22.36
5 - 25' - 27.95
6 - 30' - 33.54
7 - 35' - 39.13
8 - 40' - 22.72
9 - 45' - 50.31
17 - 85 - 95.03
So, If we apply round up round down
at 22.5 degree angles, it should be 5 squares of movement equals 4 squares on the board. BUT, the next 8 are 7 squares of movement, then 9:8, then 8:7, then 9:8, 8:7, eventually getting out of whack again, but so far down the road as to be unimportant.
at 45 degrees, it gets much simpler.
1 - 5' - 7.07
2 - 10' - 14.14
3 - 15' - 21.21
4 - 20' - 28.28
5 - 25' - 35.35
6 - 30' - 42.42
7 - 35' - 49.49
So we have to have 3 squares of movement to move 2 squares on the board, this continues out to 20'. After that, it is 4 squares of movement to move 3 squares, then it drops down to 3:2, 3:2, 4:3, 3:2, 3:2, 4:3, 3:2, 3:2, 4:3. Eventually it gets out of whack again, but at that distance, the difference is so miniscule as to be unimportant.
So remember, if your DM ignores the 3:2 ratio of diagonal movement, move diagonal whenever possible, you always pick up extra distance.
If your DM uses the 3:2 ratio of movement, always move like a knight (2 squares straight, jog one. 2 squares, jog one) for maximum distance.
There are no official rules for measuring on a grid. The grid rules will appear in the DMG as an optional rule.
Rather than doing all the calculations that captnq does, here are some simple things to think about.
Pythagorean Theorem (a^2 + b^2 = c^2) tells us that moving on a diagonal is the square root of 2.
The square root of 2 is approxamately 1.41.
If you count every other square as double (as you did in 3.x) then you are basically multiplying your diagonal squares by 1.5 and rounding down.
So we can see that counting every other square as double is REALLY REALLY close to the actual value you would get with the pythagorean theorem. It's easy to do and it's close enough to mathematically accurate to be sufficient, so why bother with anything else?
So remember, if your DM ignores the 3:2 ratio of diagonal movement, move diagonal whenever possible, you always pick up extra distance.