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I have built a 5th Level Rogue Swashbuckler who took Magic Initiate feat at Level 4 to do some extra damage with Booming Blade and Green Flame Blade.

When using a normal rapier with Booming Blade at current level on a Sneak Attack he deals: 1d8 weapon damage + Dex mod + 1d8 thunder damage and 3d6 Sneak Attack.

Do all of the dice double on a Critical Hit/"Nat 20"?

Would the total roll then be: 2d8 weapon damage + Dex mod + 2d8 thunder damage + 6d6 Sneak attack damage?

So would this be an instance that a weapon attack, a spell, and ability crits?

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Your calculation is correct

When your score a critical hit all damage dice are doubled. This also works for Paladin Smites (which are declared after the attack roll) and all other damage increasing spells/abilities (e.g. Hex, Hunter's Mark, etc.).

Critical Hits

When you score a critical hit, you get to roll extra dice for the attack's damage against the target. Roll all of the attack's damage dice twice and add them together. Then add any relevant modifiers as normal. (PHB.196)

However...

As Vylix points out, Booming Blade deals an addition 2d8 damage if the target moves before the start of your next turn. This damage is not affected by the critical hit. The same goes for Green Flame Blade, which deals fire damage equal to your spellcasting modifier to a secondary target. Critical Hits do not affect the secondary target damage.

Note: Answer is correct for 5th as given by OP. Spell damage and critical differs by class level.

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    \$\begingroup\$ The splash damage does not crit : sageadvice.eu/2017/05/24/… \$\endgroup\$ – Vylix Dec 13 '18 at 7:08
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    \$\begingroup\$ @Vylix another great example of Sage Advice going against RAW and logic \$\endgroup\$ – András Dec 13 '18 at 7:14
  • \$\begingroup\$ @andrás I think you are mistaken. Critical hits only double the damage dealt when the attack hits. Booming blade sets a conditional effect to deal damage at a later turn. No crit I know of deals extra damage after the attack is resolved. \$\endgroup\$ – Mindwin Dec 13 '18 at 11:23
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    \$\begingroup\$ @Mindwin I believe Andras was referring to Green Flame Blade not Booming Blade. \$\endgroup\$ – linksassin Dec 13 '18 at 11:37
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    \$\begingroup\$ @Mindwin They are 5th level the calculation is correct. \$\endgroup\$ – linksassin Dec 13 '18 at 11:37

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