Armor penetration

Armor Penetration and Armor Reduction  are the statistics of a Champion which allows them to counter 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.

Calculations
Armor penetration and armor reduction are applied in the following order:
 * 1) Armor Reduction
 * 2) Flat Armor Penetration
 * 3) Percentage Armor Penetration

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 reduced by the amount of 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

 * The target's armor is reduced by a percentage.
 * If the target has 30 armor and you have 20% armor pentration, the target will take damage as if it had only 24 armor.
 * 30 * (1.0 - 0.2) == 30 * 0.8 == 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 magic will lose 16.
 * Pecentage armor penetration cannot reduce a target's resistance below zero.

Examples
You have 20 armor reduction, 10 flat armor penetration, and 40% armor penetration.
 * Target A has 80 armor.
 * The 80 is reduced to 60 by your 20 armor reduction.
 * The 60 is reduced to 50 by your 10 armor penetration.
 * The 50 is reduced to 30 by your 40% armor penetration.
 * 50 * (1.0 - 0.40) == 50 * 0.6 == 30
 * Target A takes damage as if it has 30 armor.
 * Target B has 18 armor.
 * The 18 is reduced to -2 by your 20 armor reduction.
 * The -2 is not reduced by your 10 armor penetration (cannot penetrate below 0).
 * The -2 is not reduced by your 40% armor penetration (cannot penetration below 0).
 * Target B takes damage as if it has -2 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 Passive: Attacks penetrate 40% of the defender's armor. 2090 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% movespeed 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.

Masteries

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

Runes

 * Desolation runes provide flat armor penetration.

Hero Abilities

 * : Active: Kassadin charges his nether blade, gaining (7 / 15 / 25 / 38 / 50) armor penetration for 5 seconds.
 * : Passive: Olaf has (10 / 20 / 30) increased armor penetration.

Items

 * +20% Attack Speed UNIQUE Aura: Gives nearby allied Champions 20% Lifesteal, 20% Attack Speed, and 30 Health per 5 seconds regenerated. 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.

Hero 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.
 * : 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 4 seconds.
 * : 12 / 14 / 16 / 18 / 20 % Armor Reduction for 5 seconds