# What is a radius on a square grid?

So last night, my D&D group ran into an interesting question. The spell Entangle says:

Area: plants in a 40-ft radius spread.

We (like a lot of people, I think) play D&D on a square grid, where each square represents a 5'x5' area. In that context, how do you find the area when you're given a radius?

The books seem unclear, probably since the definition of a radius is fairly simple. My suggestion was to treat it the way you treat range: start at the center, and any square you can get to with 8 five-foot 'steps' is affected. Someone else thought it should be a square that was 80'x80'; that is, a 'circle' that's 80 feet across, since a lot of other 'circles' in D&D are actually treated as squares. That's a simply huge area, though -- bigger than the board we use. We wound up using a 40x40 square, since my counting would take more time and the 80x80 square was huge. 40x40 was still big enough to encompass the entire fight.

My question: how does a radius measurement (and by extension, circles in general) work on a square grid?

• Just get a strip of paper in the right length (radius) and see which squares it covers at least 50% of, when you use it to draw a circle around the centre point. Those squares are affected. – Dakeyras Dec 8 '12 at 18:03
• TL;DR version: Manhattan Circle or Euclidean Circle. – Lie Ryan Jul 4 '14 at 11:38

You can extrapolate from these spell area diagrams and the rules for determining the exact radius in squares.

Regardless of the shape of the area, you select the point where the spell originates, but otherwise you don't control which creatures or objects the spell affects. The point of origin of a spell is always a grid intersection. When determining whether a given creature is within the area of a spell, count out the distance from the point of origin in squares just as you do when moving a character or when determining the range for a ranged attack. The only difference is that instead of counting from the center of one square to the center of the next, you count from intersection to intersection.

You can count diagonally across a square, but remember that every second diagonal counts as 2 squares of distance. If the far edge of a square is within the spell's area, anything within that square is within the spell's area. If the spell's area only touches the near edge of a square, however, anything within that square is unaffected by the spell.

(The diagram and text are from Pathfinder, but these rules have not changed since 3.5.)

• +1 For pictures. This makes so much more sense visually. – IgneusJotunn Dec 8 '12 at 19:13
• +1 after reading the question, these were the first images to pop into mind here too. Well played. – LitheOhm Dec 9 '12 at 3:19
• Strictly speaking, the rules have changed between 3.5 and Pathfinder: 3.5 has you draw an actual circle, and include any square where the “majority” is within the circle. At very-large radii, this would be different from the squares you could reach using the 1-2-1-2 diagonal method. (The templates are still valid since none of them are large enough to run into such discrepancies.) – KRyan Jun 21 '17 at 19:01

The Dungeon Master’s Guide page 28 has rules for spells’ areas, and page 307 has diagrams showing circles of various radii on a square grid.

Exact text:

To employ the spell using a grid, the caster needs to designate an intersection of two lines on the grid as the center of the effect. From that intersection, it’s easy to measure a radius using the scale on the grid. If you were to draw a circle using the measurements on the grid, with the chosen intersectionat the center, then if the majority of a grid square lies within that circle, the square is a part of the spell’s area.

Thus, the squares that would actually lie inside a real circle drawn to scale, with an intersection – not a square – as the center. The Dungeon Master’s Guide specifies that the “majority” of the square needs be in such a circle.

This more-or-less works out to including any squares that could be reached by moving a distance of the radius. The first square (since you start on an intersection) is five feet; from there count the extent of how far you can get by moving a distance equal to the radius. At very large radii, this method may not exactly match the official one.

The official method is to use one of the grid layouts in the book.

The math based method can be done one of several ways. All of which give slightly different results.

• String method
• calculation method
• template method
• angled count
• range table

Now, assuming the use of 1"=5' grid maps...

# String method

Take a string with a loop at one end, put that loop over a nail or pencil, hold that over the center of the effect. Measure a straight line to the distance on the string, then sweep over to see if the center of the square or hex you need to know about is less than the range.

This works for square grid, hex grid, or gridless play.

A close variant is to whip out a ruler or tape measure, and simply measure, remembering that

5'=1"
10'=2"
15'=3"
20'=4"
25'=5"
30'=6"
35'=7"
40'=8"
etc.

# Calculation Method

This only works easily for square-grid play.

Measure orthogonally straight out to parallel to the questioned figure. This is X. Then measure over from that file, this is Y.

D=√((X * X)+(Y * Y))

if you don't have a calculator with a squareroot key... if (X*X)+(Y*Y) ≤ R*R, it's in range.

So, if range is 40', convert that to 8 squares. a guy 4 up and 3 over...
is (4*4)+(3*3)≤ 8*8 ?
16+9≤64 is true, so yes, it's in range.

# Template Method

Good for gridded or gridless play.

Make or buy clear templates in the common radii...

5' isn't needed - it affects only adjacent orthogonal squares.

10' is a 2" radius disk - find a 4" across can lid.
15' is a 3" radius disk - find a 6" across can lid, or trim down from a larger one.
20' is a 4" radius disk - find an 8" across can lid. Look for coffee cans.

30' and 40' are best made as quarter circles - they get unwieldy, being 6" and 8" radius - but can be made on a single overhead transparency sheet that way.

# Angled Count

Angled count works on both square grids

Count diagonally by 1.5's until orthogonal to the target, then by 1's to the target.

So, if the target is 6up by 5 over, I'd count 5 diagonally, and 1 orthogonally. The count would be "1.5, 3, 4.5, 6, 7.5, 8.5 inches"... which, for a 40' effect, would be out of range.

Note that angled count has an issue - it's a rough approximation, and a diagonal is √2 which isn't 1.5, it's 1.4142 or so... but it's a close enough approximation for gaming use.

# Range Table

Set up a table with ranges pre-figured for X and Y of the calculation method.

X Y: 1   2   3   4   5
1    5'  8' 16' 21' 26'
2    8' 15' 18' 24' 27'
3   16' 18' 22' 25' 30'
4   21' 24' 25' 29' 32'
5   26' 27' 30' 32' 36'


I've rounded to next whole foot after working to 1 decimal.

I found this radius example image on ENworld. Not that a "radius" on the grid leaves out several squares that would be affected if you were to use the pencil and string method.

If your really want to get hard-core, the thread in question also discusses radius templates in 3D space. http://www.enworld.org/forum/showthread.php?190797-Area-of-Effect-Size-Printable-Templates/page7