Armor

Armor is a stat shared by all units, including monsters, and buildings. Increasing armor reduces the physical damage the unit takes. Each champion begins with some armor which increases with level ( being the only exception). You can gain additional armor from abilities, items, and runes. Armor stacks additively.

Excluding whose base armor does not scale with levels, base armor ranges from   to   at level 18.

Damage reduction

 * Note: One can include the armor penetration in all the following ideas by enumerating it with a due amount of corresponding negative armor.

Incoming physical damage is multiplied by a factor based on the unit's armor:

$$\pagecolor{Black}\color{White}{\rm Damage\ multiplier}=\begin{cases}{100 \over 100+{\it Armor}}, & {\rm if\ }{\it Armor} \geq 0\\2 - {100 \over 100 - {\it Armor}}, & {\rm otherwise}\end{cases}$$

Examples:
 * 25 armor &rarr; × 0.8 incoming physical damage (20% reduction, +25% effective health).
 * 100 armor &rarr; × 0.5 incoming physical damage (50% reduction, +100% effective health).
 * -25 armor &rarr; × 1.2 incoming physical damage (20% increase, -16.67% effective health).

Stacking armor
Every point of armor requires a unit to take 1% more of its maximum health in physical damage to be killed. This is called effective health:
 * $$\pagecolor{Black}\color{White}{\rm Effective\ health} = (1 + \frac{100})\times{\rm Nominal\ health}$$


 * Example: A unit with 60 armor has 60% more of its maximum health in effective health, so if the unit has 1000 maximum health, it will take 1600 physical damage to kill it.

What this means: by definition, armor does not have diminishing returns in regard of effective hitpoints, because each point increases the unit's effective health against physical damage by 1% of its current actual health whether the unit has 10 armor or 1000 armor. However, health and armor have increasing returns with respect to each other.


 * Example: A unit starts with 1000 health and 100 armor giving it 2000 effective health. Now, it increases its nominal health from 1000 to 2000, thereby increasing its effective health from 2000 to 4000. Increasing the unit's armor by 100 at both nominal health levels would yield +1000 effective health and +2000 effective health, respectively. If we were to consider two nominal armor levels and then increase both by a static amount of health, we would see a similar increased return of effective health for the same nominal health.

Therefore, buying only armor is gold inefficient compared to buying the optimal balance of health and armor. It is important to not stack too much armor compared to health or else the effective health will not be optimal.

When a unit's armor is negative because of armor reduction or debuffs, armor has increasing returns with respect to itself. This is because negative armor cannot reduce effective health to less than 50% of actual health. A unit with -100 armor has 66.67% of nominal health (gains ) of its maximum health as effective health.

Armor as scaling
These use the champion's armor to increase the magnitude of the ability. It could involve total or bonus armor. By building armor items, you can receive more benefit and power from these abilities.

Champions

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Ways to reduce armor
Note that armor penetration and armor reduction are different.

Armor vs. health
''Note: The following information similarly applies to magic resistance. As of season six, the base equilibrium line for armor is a function:''

health(armor100)

while for magic resistance the line is a bit shifted down and less steep:

health(magic resistance100)

It can be helpful to understand the equilibrium between maximum health and armor, which is represented in the graph on the right. The equilibrium line represents the point at which your champion will have the highest effective health against that damage type, while the smaller lines represent the baseline progression for each kind of champion from level 1-18 without items. You can also see that for a somewhat brief period in the early game health is the most gold efficient purchase, however this assumes the enemy team will only have one type of damage. The more equal the distribution of physical damage/magic damage in the enemy team, the more effective will buying health be.

There are many other factors which can effect whether you should buy more armor or health, such as these key examples:
 * Unlike HP, increasing armor also makes healing more effective because it takes more effort to remove the unit's HP than it does to restore it.
 * HP helps you survive both magic damage and physical damage. Against a team with mainly burst or just low magic damage, HP can be more efficient than MR.
 * Percentage armor reduction in the enemy team tilts the optimal health:armor ratio slightly in the favor of HP.
 * Whether or not the enemy is capable of delivering true damage or percent health damage, thus reducing the value of armor and health stacking respectively.
 * The presence of resist or HP steroids built into your champion's kit, such as in or.
 * Against sustained damage life steal and healing abilities can be considered as contributing to your maximum HP (while being mostly irrelevant against burst damage).
 * The need to prioritize specific items mainly for their other qualities (regardless of whether or not they contribute towards the ideal balance between HP and resists).

Optimal efficiency (theoretical)

 * Note: Effective burst health, commonly referred to just as 'effective health', describes the amount of raw burst damage a champion can receive before dying in such a short time span that he remains unaffected by any form of health restoration (even if the actual considered damage is of sustained form). Unless champion's resists aren't reduced below zero, it will always be more than or equal to a champion's displayed health in their health bar and it can be increased by buying items with extra health, armor and magic resistance. In this article, effective health will refer to the amount of raw 'physical damage' a champion can take.

In almost all circumstances, champions will have a lot more health than armor such that the following inequality will be true: Champion-Health > Champion-Armor100.

If this inequality is true, a single point of armor will give more 'effective health' to that champion than a single point of health.

If (health < armor100), 1 point of health will give more effective health than 1 of armor.

If (healtharmor100), 1 point of health will give exactly the same amount of effective health as 1 point of armor.

Because of this relationship, theoretically, the way to get the maximum amount of effective health from a finite combination of health and armor would be to ensure that you have exactly 100 more health than armor (this is true regardless of how much health and armor you actually already have).


 * Example: Given a theoretical situation where you start off with 0 health and 0 armor and are given an arbitrary sufficient number of stat points (x ≥ 100), each of which you can either use to increase your health or armor by 1 point, the way to maximize your effective health is to add points to your health until your health(armor100)(0100)100, and then split the remaining stat points in half, spend half on your health and half on your armor.

However, this is only theoretically true if we consider both health and armor to be equally obtainable resources with simplified mechanism of skill point investment. In reality a player buys these stats for gold instead. As gold value of armor (derived from cost of basic armor item) is currently (as of season six) times higher than gold value of health (derived from cost of basic health item), we theoretically can maximize effective health represented by product of  health(armor100) with gold as input variable by satisfying the following equation: health(armor100). The graph and conclusions obtained by solving it are mentioned in the subsection below.


 * Example: Given a theoretical situation where you start off with 0 health and 0 armor and are given an arbitrary sufficient amount of gold (x ≥ ), which you can either use to increase your health or armor, the way to maximize your effective health is to add points to your health until your health(armor100)(0100)750, and then split the remaining gold in half, spend half on your health and half on your armor (as former is times cheaper than the latter, it would lead to buying  times more additional health than armor and thus naturally reaching equality in the equation above).

Now we just formulated a simple rule of preserving equilibrium (or maximum effective health):

Once equilibrium state is reached, all we need to do to preserve it is to always distribute gold equally into all involved stats for the rest of the game.

... or in our case, always 50% gold into health and 50% gold into armor.

Again this model is highly simplified and cannot be exactly applied in cases when we are buying any other item than, or  (for example if our decision-making process would involve  instead of , the above model would need to use equilibrium constant ). Even considering the purchase of different armor or health items with differing gold efficiencies (quite natural expectation under real circumstances) makes use of single constant utterly impossible. Going even further, the continuous model simplifies a discrete character of real shopping, as you cannot really buy for, so with that much gold you opt to buy either a single  or 2, drastically changing the equilibrium constant to 5.

However, thankfully to almost linear item stats' gold efficiency a player can use weakened base equilibrium condition in a form: health ≈ (armor100) safely enough to speed up decision-making. The important thing to remember is that there is no reason to hold to it too strictly.

Note: In case of magic resistance only the basic constant is slightly changed to .

This information is strongly theoretical and due to game limitations from champions' base stats, innate abilities and non-linearity of gold value of item stats (gold value of stats differs for different items or is even impossible to be objectively evaluated due to interference of unique item abilities), the real equilibrium function is too complicated to be any useful.

The complexity of this problem provides space for players' intuition to develop and demonstrate their itemization skills. If given sufficient amount of time, each player could perfectly analyze situation at any given moment when he exited the shop and tell what should he buy at that moment for available gold to maximize own effective health. The sheer impossibility of doing such thing in real time creates opportunity to develop the skill. Not only that but often choosing to maximize current effective health leads to suboptimal decision branches in the future. The summary on end game screen about type of fatal damage taken is a key part of this decision process as well.

Instead, broadly speaking, items which provide both health and armor give a very high amount of effective health against physical damage compared to items which only provide health or only provide armor. These items should be purchased when a player is seeking efficient ways to reduce the physical damage they take by a large amount. Furthermore, these items are among all available items the best ones to distribute their gold value equally among both health and armor, thus working perfectly for rule of preserving equilibrium.

Trivia

 * Armor has a gold value of (30015). This value is derived from.

Last updated: December 21, 2017, patch V7.24b
 * One of the biggest amount of armor any champion can obtain, aside from, is  (which reduces physical damage by %), being a level 18 .
 * 82 armor
 * 6
 * 6
 * 6
 * 6
 * 6


 * armor:
 * Base stats:  armor
 * Items =  = 600 armor
 * Runes =  +   = 28 armor
 * Armor Amplification = 1 + ( + ) =
 * armor = (600 + 28 + ) = armor
 * bonus =  =  bonus armor


 * bonus armor:
 * Items = = 600 armor
 * Runes = +   = 28 armor
 * Bonus Armor Amplification = 1 + ( + ) =
 * armor = (600 + 28) = bonus armor
 * bonus = +  =  bonus armor


 * armor:
 * Base stats: 82 armor
 * Items =  = 600 armor
 * Runes = +   = 28 armor
 * Buffs =  +  +  + =  armor
 * Armor Amplification = 1 +  +  +   =
 * armor = (82 + 600 + 28 + ) =  armor

, with his effectively infinite stacking, can obtain a maximum of armor off his passive alone. With the same set-up as above, he can obtain a total of about 910296.7564 armor, reducing physical damage by.