Spell Areas on a Grid

One tricky aspect of D&D is adjudicating spells which have circular areas of effect, when playing on a tactical map with a square grid. In fact, this issue of "squaring the circle" is a classic mathematical problem. There are fairly good guidelines on DMG p. 67-69, but they don't quite manage to be comprehensive enough to satisfy all players; this page presents some tools for making exact usage of these spells easier.

Using a Universal Compass

To begin with, if you're playing on a standard battlemat using wet-erase markers to draw the environment, one of the best options is to simply purchase a "universal compass" at your nearest chain office-supply store. This compass will hold a variety of writing instruments, including wet-erase markers; therefore, you can quickly draw any circle directly onto the playing surface, up to about a 40-foot radius, and use the "majority of a grid square" rule for each space (from DMG p. 69). This tool has the advantages of speed, ease-of-use, and usually not needing to first move miniatures out of the way in order to draw the area. However, it may lead to arguments between players as they try to visually determine whether the spell covers a "majority of a grid square" for particular spaces.

Using Spell Templates

A more exacting method is to use pre-printed area templates which can simply be laid down on the gaming surface, much like the area spell diagrams presented on DMG p. 68-69, clearly specifying which grid spaces are covered by an otherwise circular spell effect. The rest of this page will supply you with graphics for the most common spell area templates, which can be easily printed out and used in your own game.

Choosing Between Two Measurement Rules

Unfortunately, there is a contradiction between the rules as written, and the sample diagrams shown on DMG p. 68-69. Under "Area Spells", it's written "If you were to draw a circle using the measurements on the grid, with the chosen intersection at the center, then if the majority of a grid square lies within that circle, the square is part of the spell's area. Refer to the diagram of the sleep spell's area to see which squares are covered by the burst." However, the diagram for the 15-foot radius sleep spell does not match the "majority of a grid square" rule; actually, the next diagonal spaces in each direction should have their majority lying inside the proper 3-square radius circle, and thus there should be another 8 squares added to the sleep area of effect. (This can be precisely calculated by looking at the fractional portion of the integral Integral sqrt(3^2 - x^2) dx from 1 to 2. Since this works out to about 0.576, it shows that over 57% of each such space is actually within the spell's area of effect.)

In fact, the DMG spell diagrams are instead compliant with the rule suggested outside the rulebooks by the D&D game designers, that for counting distances, every two diagonal squares be treated as 15 feet across (5 feet for the first space, 10 feet for the second). Although this rule-of-thumb is easy to use and a good approximation (the mathematical length of two diagonal spaces is indeed sqrt(10^2 + 10^2), or over 14 feet), at larger scales it produces smaller-than-actual areas (it loses one space after about 40 feet, a second space after the 120-foot mark, and so forth).

Below, I have supplied templates which conform to each of these methods. The first method I've referred to as the "Majority Rule" (precisely computed via integral calculus, closely simulating the actual surface area covered by each spell, and complying with the written text of the DMG); the second method I've called the "Five/Ten Rule" (approximately measured, and matching the diagrams in the DMG). You'll have to decide which determination procedure you'll actually want to use in your game.

Printing Spell Templates

Here are a couple of things you should check before printing the templates below, to make sure they're produced at the proper scale and using as little paper as necessary.

Set margins at 0.25". Make sure the left and right printing margins in your browser or graphics program are set as small as possible, so it uses a full 8" across on a normal sheet of paper. This allows you to print anything up to a 20-foot scale radius using only a single sheet of paper, and with as little manual construction as possible (for larger templates, you'll still have to cut and glue several section together yourself).

Set your display to 96 dpi. These templates are designed at the Windows standard of 96 dpi (pixels, or dots-per-inch); some programs will ignore the specification in the graphics file, and simply print according to the display setting of the operating system, causing the templates to come out at the wrong scale (such as with Microsoft Paint and Netscape Navigator). If you're using one of these programs, make sure your display is at 96 dpi (you can check this in Windows by inspecting "Start->Settings->Control Panel->Display->Settings->Advanced->General->Font Size"). A more sophisticated graphics program, such as Adobe Photoshop, will print the templates at the correct scale regardless of your display settings.

Spell Templates: Radial Areas (bursts, emanations, spreads, cylinders)

The most common type of rounded area of effect is simply a circle of some predefined radius; as the rules specify for this type of spell, "the caster needs to designate an intersection of lines on the grid as the center of effect" (DMG p. 69), which makes them moderately easy to adjudicate. Note that in the case of "spreads", any distance the effect needs to travel around a corner or barrier counts against the radius length; however, as long as there are no unusual obstacles, the following templates are accurate for all types of Radial areas of effect.

Radial Areas (Centered on Intersection of Lines)
Radial AreaNo. of SpellsExample(s)Printed PagesMajority RuleFive/Ten Rule
5 ft.3faerie fire1viewview
10 ft.7sound burst1viewview
15 ft.9sleep, confusion1viewview
20 ft.8fireball, web1viewview
30 ft.10fog cloud, etc.4viewview
40 ft.3sleet storm4viewview

There is a further complication in that certain types of emanations specifically radiate from the caster or another creature, and so cannot properly be centered on an intersection of lines. Your options in this case are to (a) use a line intersection for the center anyway, and use the preceding templates (which will be off-centered from the actual recipient); or (b) use the following templates specially centered on the middle of a space instead of a line intersection. These square-centered templates are designed to duplicate the number of spaces covered by the line-intersection versions, above, as closely as possible (in fact, each covers exactly one more space than the matching radius above, due to the square in the center).

Radial Areas (Centered on Grid Square)
Radial AreaNo. of SpellsExample(s)Printed PagesMajority RuleFive/Ten Rule
10 ft.11magic circle, etc.1viewview
20 ft.7holy aura, etc.1viewview
30 ft.5prayer, holy word4viewview

Spell Templates: Group Targets (creatures within a restricted distance)

Spells of this second most common type of rounded area usually have a description such as "Targets: One creature/level, no two of which can be more than 30 ft. apart", which is a bit more complicated than it first looks. The designers may have intended to simply mean that all the creatures must fit within a 30-foot diameter circle, but that's not the same as what this language describes. For example, it is possible to triangulate three creatures, each exactly 30 feet apart from the others, but requiring at least a 35-foot diameter circle to encompass them all.

You may wish to use one of the following strategies for spells of this type: (a) just approximate them by using a circular area of the specified diameter (using any of the previous Radial templates); (b) simply use a square area of the specified length, which is very close to the exact area; or (c) use the following precise Group templates, which guarantee that no two creatures are outside the specified distance. (Note that these are respectively identical to the areas of the 10-foot-radial and 30-foot-radial areas calculated with the "Majority Rule". Even if you otherwise use the "Five/Ten Rule", you should use these larger areas to best simulate these spells.)

Group Targets
Maximum DistanceNo. of SpellsExample(s)Printed PagesEither Rule
15 ft.1magic missile1view
30 ft.50remove fear, summon1view

Spell Templates: Arced Areas (semicircles, quarter-circles, cones)

A small but important set of spells is composed of those which approximate some slice of a circle: semicircles, quarter-circles, and cones. Examples of each of these different types can be found in the diagrams on DMG p. 68-69; by the rules, they must be centered on an intersection of lines and shot either straight or diagonally along the grid. Because of this, they usually have fairly complicated and asymmetric shapes (in fact, two of the three diagrams in the DMG had to be corrected by errata; for a suggestion of a simpler alternative rule, see the appendix at the bottom of this page).

Although the rules say you should "remember to maintain a consistent number of affected squares in areas that differ on the diagonal" (DMG p. 69), it is not possible to precisely do this while maintaining consistent shapes (for example, both the diagrams for color spray and detect evil in the DMG, as illustrated, have a different number of spaces in the straight and diagonal versions). It should be acceptable if two such templates have very nearly the same number of spaces, as do the following.

Note that most "cone" spells have a variable length (based on caster level), so the applicable templates below are graduated and meant to be used with a variety of cone lengths. Significant lengths for these cones are marked on the templates by dark-gray banding; it should be simple to interpolate other lengths by just adding another set of straight or diagonal squares for each added 5 feet in length.

Arced Areas
AreaNo. of SpellsExample(s)Printed PagesMajority RuleFive/Ten Rule
Semicircle1burning hands
10 ft. radius, straight1viewview
10 ft. radius, diagonal1viewview
1/4-circle8detect evil, etc.
60 ft. radius, straight4viewview
60 ft. radius, diagonal4viewview
Cone5color spray, shout
variable length, straight4viewview
variable length, diagonal4viewview

Spell Templates: Meteor Swarms

For the final category, I've set the meteor swarm spell aside by itself; as a 9th-level wizard spell, its use is very rare and you may never really need to have a special template on hand for it. However, when it does get used, it makes some fairly complicated and severe patterns, and you may wish to be very exacting with it (see PH p. 227); therefore, if you're running a character who gains access to this spell, it's probably smart to print these templates out and have them on hand whenever a meteor swarm does get used. (Note that this spell differs a bit from other area-type spells, in that it's targeted at a square center instead of a line intersection, according to the diagrams in the PH.)

The color codes for these templates are as follows: light gray = one sphere area; dark gray = two sphere area; black = three sphere area; inner white = four sphere area. Note also that I've made one minor change from the diagrams on PH p. 227 -- technically, the "large sphere" patterns should have 4 two-sphere-areas on each limb, and so I've added those missing spaces to the templates below. (If you wish to switch back to the PH charts, I'm sure you'll find it easy to see the spaces I added and color them back with any common paint program.)

Meteor Swarms
PatternPrinted PagesEither Rule
small sphere, diamond-within-box1view
small sphere, box-within-diamond2view
large sphere, box2view
large sphere, diamond4view

Appendix

For a presentation of a simpler, alternative spell area rule than that presented in the DMG, see here.