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The setting is gladiator pit, warlock vs two weapon fighter. The warlock has cast plant growth and covered the ground in thick vegetation. Fighter becomes frustrated by his inability to move quickly and throws alchemist’s fire on the ground. Fire soon spreads cutting arena in half. Fighter then begins dousing one 5-foot section per turn, trying to move through the ensuing burning difficult terrain.

If a character is surrounded by flames—be it 5 feet or 30 feet radius—but the square they are in is not burning, what damage would they take? Or would it be a constitution issue and cause exhaustion?

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    \$\begingroup\$ Are you using alchemists fire from the books or something homebrew? Your spreading mechanic sounds homebrew, so double checking. \$\endgroup\$ – NautArch Sep 6 at 22:57
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    \$\begingroup\$ And are you asking for mechanics for how folks have done this/could do this in the rules or are you asking for confirmation if there is a mechanic that you should be using with alchemists fire. \$\endgroup\$ – NautArch Sep 7 at 12:35
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    \$\begingroup\$ Welcome to RPG.SE! Take the tour if you haven't already, and check out the help center for more guidance. \$\endgroup\$ – V2Blast Sep 8 at 5:39
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Wilderness Survival

The closest 5e has to a written rule about enduring high temperatures without actually being on fire is from the Adventure Environments section of the DMG (page 110):

When the temperature is at or above 100 degrees Fahrenheit, a creature exposed to the heat and without access to drinkable water must succeed on a Constitution saving throw at the end of each hour or gain one level of exhaustion. The DC is 5 for the first hour and increases by 1 for each additional hour. Creatures wearing medium or heavy armor, or who are clad in heavy clothing, have disadvantage on the saving throw. Creatures with resistance or immunity to fire damage automatically succeed on the saving throw, as do creatures naturally adapted to hot climates.

But obviously, this rule is really about long-term survival in the wilderness in a harsh environment, at temperatures which fall a bit short of effectively sitting in the middle of a forest fire, and is operating on an hourly scale which isn't relevant to your described scenario. In that arena, the danger to the fighter isn't that they're going to get dehydrated and exhausted; it's that they're breathing in smoke and at risk of getting burned just by the radiating heat from the fire, so it makes sense for this to be an environment which can actually cause damage to the fighter.

Improvising Damage

In the absence of specific rules, if you want to represent the danger of the fire, you will have to come up with some ruling as the DM. The DMG, on page 249, has guidelines on Improvising Damage based on the circumstances. It suggests, for instance, 1d10 for "being burned by coals", or 2d10 for "stumbling into a fire pit".

I would personally be inclined to judge that in the scenario described in your question, being up close amidst the fire is perilous enough to merit 1d10 damage at the start of a character's turn (surrounded by but not in the flames), but actually stepping into the fire would do 2d10. This would be quite dangerous for low-level heroes without many hit points but higher level characters would find themselves readily leaping through flames without much fear, as great heroes are wont to do.

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It's DM fiat

Since there is nothing specific about being next to a fire, other than for telling stories, in the PHB or DMG, we would normally consult spell that have similar effects.

The first example would be Wall of Fire:

You create a wall of fire on a solid surface within range...

When the wall appears, each creature within its area must make a Dexterity saving throw. On a failed save, a creature takes 5d8 fire damage, or half as much damage on a successful save.

One side of the wall, selected by you when you cast this spell, deals 5d8 fire damage to each creature that ends its turn within 10 feet of that side or inside the wall.

So it's a big fire anything on the hot side takes damage. Well, this isn't magical, so there would be no "not hot" side. But it gives a space of 10ft to take damage.

Next up is Flaming Sphere:

A 5-foot-diameter sphere of fire appears in an unoccupied space of your choice within range and lasts for the duration. Any creature that ends its turn within 5 feet of the sphere must make a Dexterity saving throw. The creature takes 2d6 fire damage on a failed save, or half as much damage on a successful one.

So now it's anyone within 5ft of the source. A little more realistic in that heat goes everywhere. That should be the answer, right? Not quite. There is also a cantrip called Create Bonfire:

You create a bonfire on ground that you can see within range. Until the spell ends, the magic bonfire fills a 5-foot cube. Any creature in the bonfire's space when you cast the spell must succeed on a Dexterity saving throw or take 1d8 fire damage.

The bonfire ignites flammable objects in its area that aren't being worn or carried.

So a bonfire only damages people and object in its own space and doesn't do any damage to the squares next to it.

So it could be anything from 10ft to 5ft to it doesn't matter. It depends on how the DM sees the fire raging.

Then you can add on heat exhaustion rules...

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You could simply make the external ring of fire work in a similar way to a Wall of Fire spell: the fire itself is so strong that merely by standing at some distance you take some damage.

If the ring of flames is "normal" size, not "epic" size, then no per-round fire damage, just apply the extreme heat rules.

I have several times in fact used this exact method to simulate bonfire damage or even "entering a house on fire to go save a kid" damage. In all my campaigns, I always tell my players that magic is not 100% separate from reality but actually what "mundane" reality is actually made of. For example, all fire, whether from a candle or a fireball of from the Elemental Plane of Fire, is, at its core, of the same "fiery magical essence". This is a good rationale that I give my players for when I spontaneously tweak magic according to the environment, or use magical rules for mundane effects from the environment.

I call it the "If it looks like a duck, quacks like one, and swims like one, then you might as well treat it just like one" philosophy. Yes, I also used flaming sphere, but only for small or for less intense fires. I mainly look not at how big but at how high a natural mundane fire goes to determine how "strong" it is.

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    \$\begingroup\$ Of note: Flaming Sphere is a little smaller than Wall of Fire and also affects adjacent creatures/spaces. \$\endgroup\$ – Rykara Sep 7 at 0:58
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If the temperature is over 100 degrees Fahrenheit (about 37.8 degrees Celsius) you can use the rules for extreme heat from the Dungeon Master's Guide. But nothing explicitly says they would take damage from it if they're not actually IN the fire.

Extreme Heat

When the temperature is at or above 100 degrees Fahrenheit, a creature exposed to the heat and without access to drinkable water must succeed on a Constitution saving throw at the end of each hour or gain one level of exhaustion. The DC is 5 for the first hour and increases by 1 for each additional hour. Creatures wearing medium or heavy armor, or who are clad in heavy clothing, have disadvantage on the saving throw. Creatures with resistance or immunity to fire damage automatically succeed on the saving throw, as do creatures naturally adapted to hot climates.

The DMG also has rules for improvising damage and suggests that stumbling into a fire pit should be 2d10 damage which makes sense if your character actually stumbles into the space that's on fire. But if they aren't in the fire themselves I wouldn't necessarily deal damage. Although if you or the DM feel it's warranted perhaps make the character make a dex save and on a fail take 1d10 damage for stepping on a still hot coal or being burned by embers nearby (no damage on a successful save). Credit to Carcer for mentioning the improvised damage rules in their answer.

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