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This page explains how damage works in X-COM: Apocalypse.
Projectile weapons, if they hit, do 50-150% of rated damage to whatever they hit. Explosions do not roll damage - what it says is what you take (after adjustment for distance from epicentre).
What do projectile explosives (Auto Cannon, Heavy Launcher, etc.) do? Both. First the shot hits something, doing 50-150% of rated damage to whatever it hit, and then the explosion occurs, doing 100% of rated damage at the epicentre. This is why the Dimension Missile can pop shields in one shot despite "only" doing 110 damage; if it hits you, then you're actually taking 55-165 + 110 (whereas a Vortex Mine will always inflict 150 at ground zero, not enough to pop a shield). Note that these are separate instances of damage, so armour applies to each "hit" separately and a soldier can potentially take two critical wounds from being shot with one (one from the impact and one from the explosion).
Explosion power is reduced, per tile away from ground zero, by the "depletion rate" stat (which is coded for each weapon in the executable, alongside the various other parameters that define a weapon). The visual size of the fireball is not related to how far damage spreads (the latter is usually more). Almost all explosives have very similar safe distances, regardless of their power, because larger explosives have higher depletion rates.
|Explosive||Rated Damage||Depletion Rate||Safe Distance|
|Auto Cannon HE||33||4||9|
|Heavy Launcher HE||90||11||9|
Of course, since explosion damage is reduced by armour, you can frequently stand *much* closer than this without taking actual health damage (though you'll still take armour damage).
Damage to (unshielded) units
The apparent sequence of events for a unit being shot or hit by an explosion is as follows:
1) Damage modifier for armour type is applied (for units with innate armour, use unit's damage modifier).
2) Armour takes armour damage from the shot (if it is not Stun). Armour damage = (shot damage * 3 / armour rating). Armour rating is not yet recalculated.
3) Bleedthrough damage is calculated. Bleedthrough = shot damage - (original) armour rating.
4) Damage modifier for unit type is applied to bleedthrough damage (skip this for units with innate armour).
5) Final damage is applied to unit's health (or to unit's stun level if it is Stun).
6) Critical wound test: if final damage > (max health/8), chance of critical wound is (final damage/max health).
7) Armour rating is reduced by the amount of armour damage taken (in the vanilla game or for innate armour, at least; you can mess around with the ratio for item-based armour if you edit item stats).