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The Feat Channel Ray allows you to make channel Energy as a Ray attack. This of course means that there is an attack roll involved in resolving the attack.

You must succeed at a ranged touch attack to hit an unwilling target; your target is then affected by the channeled energy as normal and receives a saving throw.

Would this allow the cleric to critical hit with Channel Energy? And if so, would this extend to any effect that converts an area of effect into a single target effect requiring an attack roll?

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Yes, you can critically hit with ray attacks (and anything involving an attack roll)

Using Channel Ray means your Channel Energy ability is a ray effect rather than a burst:

When you channel energy, you can project a ray from your holy symbol instead of creating a burst.

And you can critically hit with ray effects, as per the general rules about magical effect targeting:

If a ray spell deals damage, you can score a critical hit just as if it were a weapon. A ray spell threatens a critical hit on a natural roll of 20 and deals double damage on a successful critical hit.

The verbiage in this section does often refer to spells, but the rules are valid for any ray effects regardless of whether it's produced by a spell or other ability. Ray effects are generally treated as a kind of weapon for most purposes by the rules - including the option to take feats like Weapon Focus (Ray) and Improved Critical (Ray).

It's also explicitly confirmed that you can critically hit with close range touch effects:

A touch spell threatens a critical hit on a natural roll of 20 and deals double damage on a successful critical hit.

And of course this is all in keeping with the general rule about critical hits:

When you make an attack roll and get a natural 20 (the d20 shows 20), you hit regardless of your target’s Armor Class, and you have scored a “threat,” meaning the hit might be a critical hit (or “crit”).

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A spell that requires an attack roll can score a critical hit. A spell attack that requires no attack roll cannot score a critical hit. If a spell causes ability damage or drain (see Special Abilities), the damage or drain is doubled on a critical hit.

If you're making an attack roll in order to hit the target, you have the potential to critically hit. However, this is only relevant if your effect does hit point or ability damage/drain; otherwise, a critical hit produces no extra special effect.

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Yes.

From the "Critical Hit" section of the Attack action (CRB, p.184):

When you make an attack roll and get a natural 20 (the d20 shows 20), you hit regardless of your target’s Armor Class, and you have scored a “threat,” meaning the hit might be a critical hit (or “crit”). To find out if it’s a critical hit, you immediately make an attempt to “confirm” the critical hit—another attack roll with all the same modifiers as the attack roll you just made. If the confirmation roll also results in a hit against the target’s AC, your original hit is a critical hit. (The critical roll just needs to hit to give you a crit, it doesn’t need to come up 20 again.) If the confirmation roll is a miss, then your hit is just a regular hit.

A critical hit means that you roll your damage more than once, with all your usual bonuses, and add the rolls together. Unless otherwise specified, the threat range for a critical hit on an attack roll is 20, and the multiplier is ×2.

Exception: Precision damage (such as from a rogue’s sneak attack class feature) and additional damage dice from special weapon qualities (such as flaming) are not multiplied when you score a critical hit.

The text never specifies the source of the attack, only that you have to make an attack roll. Therefore, anything that requires an attack roll can crit and cause at least double the damage dice.

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