Armor

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

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

Damage reduction

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

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

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

Examples:
 *  &rarr; (20% reduction, ).
 *  &rarr; (50% reduction, ).
 *  &rarr; (20% increase, ).

Stacking armor
Every point of requires a unit to take 1% more of its maximum health in  to be killed. This is called .
 * Example: A unit with has 60% more of its  in, so if the unit has , it will take  to kill it.

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


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

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

When a unit's is negative because of  or debuffs,  has increasing returns with respect to itself. This is because negative cannot reduce  to less than. A unit with has  (gains ) of its  as.

Optimal efficiency (theoretical)

 * Note:, 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 (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  in their  bar and it can be increased by buying items with extra ,  and . In this article,  will refer to the amount of raw  a champion can take.''

In almost all circumstances, champions will have a lot more than  such that the following inequality will be true:  > 100.

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

If ( < 100), 1 point of will give more  than 1 of.

If (100), 1 point of will give exactly the same amount of  as 1 point of.

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


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

However, this is only theoretically true if we consider both and  to be equally obtainable resources with simplified mechanism of skill point investment. In reality a player buys these stats for instead. As of  (derived from cost of basic armor item) is currently (as of season six)  times higher than  of  (derived from cost of basic health item), we theoretically can maximize  represented by product of  (100) with  as input variable by satisfying the following equation: (100). The graph and conclusions obtained by solving it are mentioned in the subsection below.


 * Example: Given a theoretical situation where you start off with and  and are given an arbitrary sufficient amount of  (x ≥ ), which you can either use to increase your  or, the way to maximize your  is to add points to your  until your (100)(0100)750, and then split the remaining  in half, spend half on your  and half on your  (as former is  times cheaper than the latter, it would lead to buying  times more additional  than  and thus naturally reaching equality in the equation above).

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

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

... or in our case, always into  and  into.

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 ' or ' items with differing (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  you opt to buy either a single  or 2, drastically changing the equilibrium constant to .

However, thankfully to almost linear item stats' a player can use weakened base equilibrium condition in a form:  ≈ (100) 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 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 of item stats ( 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. 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 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 and  give a very high amount of  against  compared to items which only provide  or only provide. These items should be purchased when a player is seeking efficient ways to reduce the they take by a large amount. Furthermore, these items are among all available items the best ones to distribute their equally among both  and, thus working perfectly for rule of preserving equilibrium.

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

Champions

 * grants equal to   to himself and the target ally for 3 seconds.
 * grants to herself equal to   for 3 seconds, deals damage after that time to units around her, and retains the defensive buff for an additional 3 seconds if any enemy is struck by the blast.
 * passively grants to himself equal to.
 * deals   splash  per basic attack in a 225-radius.
 * deals.
 * passively grants to herself equal to, increased to  while below.
 * grants to himself equal to.
 * increases his by  for 6 seconds, and deals    to enemies who basic attack him.
 * grants his next two basic attack equal to   after using an ability.
 * passively grants himself and his Bastion-marked as.
 * deals.
 * steals instantly . He continues to drain an additional over 4 second before decaying 4 seconds after the drain completes.

Champion abilities
Note: Only the buff effect of these abilities is shown here, to read more information on each of these abilities, follow the link on each of them.
 * allows her to enter an egg-state for up to 6 seconds upon reaching . While in this state, she will receive an modifier of.
 * increases by   to himself and the target ally for 3 seconds.
 * increases an allied champion's by  for 4 seconds.
 * increases his by (2level) when transform into.
 * permanently gains every time he kills an enemy, up to a maximum of 30.
 * increases his by  for 4 seconds, stacking up to 8 times for a maximum of.
 * increases his by.
 * increases his by  for 4 seconds.
 * increases her by   for 3 seconds, deals damage after that time to units around her, and retains the defensive buff for an additional 3 seconds if any enemy is struck by the blast.
 * passively grants to himself equal to.
 * increases his by  upon activation and an additional   per second for 15 seconds up to a maximum of.
 * passively increases his by  when he is not under the effects of.
 * increases an allied champion's by  for as long as the ball is attached to them.
 * increases his by  for 6 seconds, and deals    to enemies who basic attack him.
 * grants her each times she damages an enemy with an ability or attack for 2 seconds. Subsequent damage will increase the duration by 2 seconds up to a cap of 8 seconds, after where damage will refresh the duration.
 * passively increases her by 5 +5 per slain elemental dragon.
 * increases his by  for 25 seconds.
 * passively grants to himself and a linked ally.
 * passively generates souls for him to collect, with each soul collected granting permanently.
 * steals instantly . He continues to drain an additional over 4 second before decaying 4 seconds after the drain completes.
 * passively grants him for each nearby enemy champion.
 * increases his by  for each enemy champion hit for 8 seconds.

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

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

(100)

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

(100)

It can be helpful to understand the equilibrium between and, 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 is the most  purchase, however this assumes the enemy team will only have one type of damage. The more equal the distribution of / in the enemy team, the more effective will buying be.

There are many other factors which can effect whether you should buy more or, such as these key examples:
 * Unlike, increasing also makes healing more effective because it takes more effort to remove the unit's  than it does to restore it.
 * helps you survive both and . Against a team with mainly burst or just low,  can be more efficient than.
 * in the enemy team tilts the optimal : ratio slightly in the favor of.
 * Whether or not the enemy is capable of delivering true damage or damage, thus reducing the value of  and  stacking respectively.
 * The presence of resist or 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 (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 and resists).

Trivia
(Last updated December 25, 2016 on patch 6.24)

Update reminder: mastery added Preseason 7 , also if and how  stacks with  e.g.  Etc.


 * has a of  (30015). This value is derived from.
 * One of the biggest amount of any champion can obtain, aside from, is  (which reduces  by %), being a level 18.


 * level 18
 * Base stats: 
 * Runes:
 * 9
 * 9
 * 9
 * 3
 * Masteries:
 * 5 points in
 * 1 point in
 * 5 points in
 * 5 points in
 * Items:
 * 6
 * Buffs:
 * (this mastery can not stack)
 * (this mastery can not stack)
 * (this mastery can not stack)
 * (this mastery can not stack)
 * (this mastery can not stack)


 * Relevant mathematics:
 * Base stats: 
 * Items
 * Runes
 * Mast. & Buffs,
 * Bonus Armor Amplification,
 * (((600158))36)
 * bonus
 * bonus
 * bonus


 * Base stats: 
 * Items
 * Runes
 * Mast. & Buffs,
 * Bonus Armor Amplification,
 * ' (((600158))+36)'
 * bonus25
 * bonus25
 * bonus
 * bonus


 * Base stats: 
 * Items
 * Runes
 * Mast. & Buffs
 * Bonus Armor Amplification,
 * Armor Amplification
 * (((((600)))+36))
 * (((((600)))+36))


 * Having an enemy with the same setup use  on  will yield a total of . This is the highest possible finite amount of  and reduces  by %.


 * bonus
 * bonus
 * bonus


 * Base stats: 
 * Items
 * Runes
 * Mast. & Buffs
 * Bonus Armor Amplification,
 * (((600))36)
 * Bonus Armor Amplification,
 * (((600))36)

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