Armor penetration

Armor Penetration is the statistic of a Champion which allows their physical attacks (autoattacks and physical damage from abilities) to ignore some or all of a target's armor. All champions have 0 base armor penetration but can increase it with items, runes or abilities for a few champions.

Armor Reduction is the effect of some champions' abilities that reduce the armor of the target unit for a few seconds. This allow ally units to benefit from the reduction of armor.

Calculations
When attacking, armor penetration and armor reduction are applied to the target champion's armor in the following order:
 * 1) Percentage Armor Reduction
 * 2) Flat Armor Reduction
 * 3) Flat Armor Penetration
 * 4) Percentage Armor Penetration

Percentage Armor Reduction

 * The target's armor is reduced by a percentage.
 * If the target has 30 armor and you have 20% armor reduction, the target will take damage as if it had only 24 armor.
 * 30 - (30 * 0.2) == 30 - 6 == 24
 * Percentage armor reduction has a bigger impact on targets with higher resistance.
 * For 40% reduction, a target with 200 armor will lose 80.
 * For 40% reduction, a target with 40 armor will lose 16.
 * Percentage armor reduction can reduce a target's resistance below zero, if stacked to pass 100%.
 * Note*: If one person were to stack a percentage reduction value past 100% it is possible to reduce armour into the negatives. But that situation is impossible and if it caused by multiple champions the reduction is calculated separately. Therefore if one champion has 50% reduction and another has 60%, the first percentage calculated would take effect on said opposing champions armour, while the second reduction will take its percentage reduction from the remaining armour. Therefore it is impossible for said armour to be reduced to/beyond 0.
 * Note*: If one person were to stack a percentage reduction value past 100% it is possible to reduce armour into the negatives. But that situation is impossible and if it caused by multiple champions the reduction is calculated separately. Therefore if one champion has 50% reduction and another has 60%, the first percentage calculated would take effect on said opposing champions armour, while the second reduction will take its percentage reduction from the remaining armour. Therefore it is impossible for said armour to be reduced to/beyond 0.

Flat Armor Reduction

 * The target's armor is reduced by a certain amount.
 * If the target has 30 armor and you have 20 armor reduction, the target will take damage as if it had only 10 armor.
 * 30 - 20 == 10
 * Armor reduction can reduce a target's resistance below zero.
 * The target will take additional bonus physical damage with negative armor.

Flat Armor Penetration

 * The target's armor is ignored by the amount of the attacking champion's armor penetration.
 * If the target has 30 armor and you have 10 armor pen, the target will take damage as if it had only 20 armor.
 * 30 - 10 == 20
 * Flat armor penetration cannot reduce a target's armor below zero.

Percentage Armor Penetration

 * A percentage of the target's armor is ignored by the amount of the attacking champion's armor penetration.
 * If the target has 30 armor and you have 20% armor penetration, the target will take damage as if it had only 24 armor.
 * 30 - (30 * 0.2) == 30 - 6 == 24
 * Percentage armor penetration has a bigger impact on targets with higher resistance.
 * For 40% penetration, a target with 200 armor will lose 80.
 * For 40% penetration, a target with 40 armor will lose 16.
 * Pecentage armor penetration cannot reduce a target's resistance below zero.

Examples
You have 30% armor reduction, 20 flat armor reduction, 10 flat armor penetration, and 40% armor penetration.
 * Target A has 80 armor.
 * The 80 is reduced to 56 by your 30% armor reduction.
 * The 56 is reduced to 36 by your 20 armor reduction.
 * The 36 is reduced to 26 by your 10 armor penetration.
 * The 26 is reduced to 15.6 by your 40% armor penetration.
 * 26 - (26 * 0.40) == 26 - 10.4 == 15.6
 * Target A takes damage as if it has 15.6 armor.
 * Target B has 18 armor.
 * The 18 is reducted to 12.6 by your 30% armor reduction.
 * The 12.6 is reduced to -7.4 by your 20 armor reduction.
 * The -7.4 is not reduced by your 10 armor penetration (cannot penetrate below 0).
 * The -7.4 is not reduced by your 40% armor penetration (cannot penetrate below 0).
 * Target B takes damage as if it has -7.4 armor.

Negative Armor
If a target has negative armor, the amount of bonus damage is equal to the amount of damage reduction caused by that positive amount of armor. This sounds confusing and is better illustrated with an example.
 * If a target has 10 armor, they take 9.1% reduced damage (formula from Armor).
 * Therefore a target with -10 armor would take 9.1% bonus damage.
 * So if you attack for 100 physical damage on the target, it would take 109.1 damage instead.
 * If a target has 20 armor, they would take 16.7% reduced damage.
 * Therefore a target with -20 armor would take 16.7% bonus damage.
 * So if you attack for 250 physical damage on the target, it would take 291.7 damage instead.

Items

 * : +40 Attack Damage UNIQUE: +40% Armor Penetration 2290 Gold.
 * : +60% Attack Speed UNIQUE Passive: Every fourth attack deals 100 extra Magic Damage. UNIQUE Active: Your attacks cannot be dodged and you gain 30 Armor Penetration for 8 sec. 1970 Gold.
 * : +25 Attack Damage UNIQUE Passive: Reduces ability cooldowns by 10%, +15 Armor Penetration. 1337 Gold.
 * : +30 Attack Damage +15% Critical Chance UNIQUE Passive: +20 Armor Penetration, 15% Cooldown Reduction. Unique Active: You gain +20% movement speed and +50% attack speed for 4 seconds. Attacking enemy units with melee attacks increases the duration by 2 to a maximum of 8 seconds. 60 second cooldown. 2687 Gold.

Hero Abilities

 * : Passive: Olaf has (10 / 20 / 30) increased armor penetration.

Masteries

 * : Ignores (2 / 4 / 6) of your enemy's armor on attack.

Runes

 * runes provide flat armor penetration.

Items

 * : +20% Attack Speed UNIQUE Aura: Gives nearby allied Champions 20% Lifesteal, 20% Attack Speed, and 30 Hp5. Reduces the Armor of nearby enemy champions by 20. 2550 Gold.
 * : +55 Attack Damage +30% Attack Speed UNIQUE Passive: Physical attacks reduce your target's Armor by 15 for 5 seconds, (maximum 3 stacks). 3065 Gold.

Champion Abilities

 * : 1 / 2 / 3 / 4 / 5 Armor Reduction each hit (max 6-10 stacks) for 2 seconds.
 * : 10 / 14 / 18 / 22 / 26 Armor Reduction for 5 seconds.
 * : While Ranks 3 and 4, the turrets are green and their attacks reduce armor and magic resist by 1 per hit stacking up to 50 times.
 * : 10 / 14 / 18 / 22 / 26 % Armor Reduction for 3 seconds.
 * : 15 / 20 / 25 / 30 / 35 Armor Reduction for 5 seconds.
 * : 5 / 10 / 15 / 20 / 25 Armor Reduction for 4 seconds.
 * : 20 / 25 / 30 / 35 / 40 Armor Reduction for 7 seconds.
 * : 20 / 25 / 30 / 35 / 40 % Armor Reduction for 12 seconds.
 * : 10 / 15 / 20 / 25 / 30 Armor Reduction for 1 / 1.5 / 2 / 2.5 / 3 seconds.
 * : While casting with over 50 rage:15 / 17.5 / 20 / 22.5 / 25 % Armor Reduction for 4 seconds.
 * : 10 / 15 / 20 / 25 / 30 Armor Reduction for 4 seconds.
 * : 15 / 20 / 25 % Armor Reduction upfront, doubles over the next 6 seconds.
 * : 12 / 14 / 16 / 18 / 20 % Armor Reduction for 5 seconds.