The Maths Behind % Armor Penetration, and Why You Should Build an LDR if They Have a *Single* Tank

Hello, the topic of armor penetration has recently come up, and the discussion around it devolved quite rapidly as it became extremely apparent that basically no one understand how % armor penetration works, or just how ludicrously powerful it is, even against the base armor of champions, let alone anyone actually building or, heaven forbid, trying to stack armor items. To rectify this situation, and by request, I have decided to explain for you all just how % armor penetration works, and I will attempt to elucidate just how powerful this often overlooked stat is. To start with, let's consider the quintessential armor penetration item, Lord Dominik's Regard ({{item:3036}}), the spiritual successor to the old Last Whisper; not *this* LW {{item:3035}}, but the item it used to be before they made it into a base item with two upgrade paths. In fact, the current LDR offers nearly exactly the same stats as the former LW, except for the fact that it has 5 additional Attack Damage, and comes at a 500g higher cost, albeit with a vastly improved build path. This makes the item both easier, and nicer to build than before, as one needs to invest a paltry 1400g to start getting their % armor pen, and their final item is even more slot efficient than ever before. As a result, purchasing an early LW, and upgrading it as the game progresses is now easier and feels much better. So, with that established, lets five into just what an LDR actually gives us, courtesy of the LoLwiki: > Lord Dominik's Regards > Stats > +45 attack damage > Passive > Unique – Last Whisper: +35% total armor penetration. Alright, so it gives a flat 45 AD, worth a little over 1,500g, (1,575g to be precise) and it provides the unique passive of 35% ***total*** armor penetration. It costs 2,800g, however, so initially it may seem unappealing due to only being 56.25% gold efficient one might think . . . BUT WAIT! That's only against a target with literally 0 armor. Unless you're getting it within the first 6 minutes of the game and hitting minions, no target will ever actually have 0 armor though. Its important to notice that it is indeed ***total armor penetration*** instead of *bonus* armor penetration. Now, we have the extremely easy task of finding out when this item becomes cost efficient. Each point of armor penetrated denies the target 20g worth of armor, and each point of flat armor penetration costs 22.5g. For every 2.9 armor the target has, LDR's Last Whisper Unique passive will remove 1.015 armor. Using the lowest of these numbers, *and* rounding down 1.015 -> 1, we find that LDR becomes over 100% gold efficient against 89 base armor, and reaches efficiency against 88 armor targets. 88 armor may sound like a lot initially, but let's look at how many champions have that much, or more ***base armor*** > {{champion:82}} > {{champion:17}} > {{champion:203}} > {{champion:26}} > {{champion:518}} > {{champion:34}} (especially if Eggnivia) > {{champion:9}} > {{champion:35}} > {{champion:91}} > {{champion:25}} > {{champion:68}} > {{champion:43}} > {{champion:50}} > {{champion:112}} > {{champion:104}} > {{champion:117}} > {{champion:238}} > {{champion:20}} > {{champion:131}} > {{champion:154}} (look, our first tank!) > {{champion:114}} > {{champion:40}} (what the actual fuck, Janna has higher base armor than ZAC?!?) > {{champion:85}} > {{champion:266}} > {{champion:75}} > {{champion:62}} > {{champion:141}} > {{champion:5}} > {{champion:106}} > {{champion:102}} > {{champion:59}} > {{champion:56}} > {{champion:36}} > {{champion:64}} > {{champion:79}} > {{champion:27}} > {{champion:28}} > {{champion:16}} (why the fuck does Soraka also have so much base armor?!?) > {{champion:421}} > {{champion:267}} (okay RIOT wtf) > {{champion:78}} (our second tank, who also gets passive %increased resistances) > {{champion:31}} > {{champion:32}} (Ayy, more tanks) > {{champion:44}} (also gets %increased armor) > {{champion:254}} > {{champion:497}} (? Seriously, RIOT, why the fuck do enchanters have way too much armor?) > {{champion:420}} > {{champion:164}} > {{champion:54}} (third highest %armor increased steroid in game, actually gets two of them) > {{champion:77}} > {{champion:432}} (okay what the actual fuck) > {{champion:72}} > {{champion:111}} > {{champion:58}} > {{champion:240}} > {{champion:80}} > {{champion:12}} > {{champion:516}} > {{champion:120}} > {{champion:113}} ( %increased armor steroid) > {{champion:223}} > {{champion:57}} > {{champion:83}} > {{champion:122}} > {{champion:89}} > {{champion:6}} > {{champion:33}} (100% increased total armor steroid) > {{champion:53}} > {{champion:201}} (also gets a %resistance steroid) > {{champion:555}} (What the actual fuck, RIOT, why does an assassin need 130 base armor, and 15 more armor than BRAUM and the highest base HP in the game by several hundred) Well, damn, that actually comes out to over 65 different champions, or roughly 46% of the ***entire League of Legends roster*** Again, against every single one of these, yes even Kindred, Neeko and Teemo, LDR is beyond cost effective with them having no additional armor from any source, be it runes, other champs abilities or their items. ~ Let's just take a minute to let that sink in. This, by the way, is not true for a single other % penetration item in the entire game. Even though {{item:3135}} grants a seemingly higher amount of AP at 70, and higher amounts of magic penetration at 40%, the AP is worth less and is a comparatively smaller damage increase than the 45 AD on {{item:3036}} . Furthermore, do to vastly lower base magic resist values, it is ***NOT*** cost effective against ***any target from their base stats*** In fact, it requires a higher amount of MR to break even than {{item:3036}} requires of armor. You need over 91 MR for {{item:3135}} to become efficient, which requires even the tankiest of champions to purchase at least ***30 additional magic resist*** and if they are not literally only Talon (who for some reason has more MR than any other champion despite being a fucking hypermobile assassin) or Pyke and Sylas (why do these two also have more MR than standard for no reason, on top of far more health than they should have for their respective classes? RIOTs just giving their new champions outstated base stats for no fucking reason) who have a base 57 MR at 18, you'll either have scaling MR and so 53.4 MR at level 18, which would require purchasing an additional 38 MR for {{item:3135}} to become efficient, or you get .5MR/level and so have 38.5 MR at 18, and need to almost triple your MR, purchasing at least 52.5 MR for {{item:3135}} to become efficient. ~ Back to the topic at hand, and our over stated friend {{item:3036}}. Remembering that it gets between 100-115% gold efficiency just from base stats against the tankiest 46% of the roster, how much armor does a squishy champion need to purchase for it to be effective against them? Well, taking the lowest armor champion in the game, {{champion:38}} at 66.6 armor and {{champion:61}} at 68, we find that they need barely 20 armor for {{item:3036}} to become 100%+ efficient. This means that even the squishiest champions in the game with so much as a {{item:3047}} are now optimal targets for {{item:3036}}. ~ You picked your jaw off the floor, and are done reeling in your chair by now, I take it. Yes, if you see any armor item at all, on any champion in the fucking game, {{item:3036}} is a great purchase. That's it. Its purchase condition is literally the presence of so much as a {{item:1029}} in the enemies inventory, or them picking one of 65+ champions. Alright, but how crazily powerful can this fucker get? Well, strap yourself in, because it can easily become the ***far and away most cost efficient item in the entire game, with NOTHING even coming close if they have a tank*** Now, it would be easy to go straight to a 5-6 item {{champion:33}} in Defensive Ball Curl, with 800-900 armor, where he's losing 5,600-6,300g worth of stats, or literally between 200% the entire cost of {{item:3036}} and 225% cost efficient, again ***from its passive a-fucking-lone*** *before* we factor in the 1575g worth of raw AD (which is further multiplied in value by any AD ratios, AS/Crit/Crit Damage) making the item between 256.25%-281.25% cost effective, which is ***BEYOND FUCKING RIDICULOUS*** but not even the worst case possible. ~ Instead, let's look at an average tank with an early, then mid game build, or just a few armor components, then items. According to League of Graphs, in NA at least, in Plat+ the most played tank is a support {{champion:111}} so I will use him as our example. Additionally, Naut is nice as he has no resistance steroids (unless he gets one from runes, but his most common rune path has Bone Plating instead) to further complicate matters. Since he's a support, and we're looking at mid game, for the first example he will be level 9, with only {{item:3047}} and {{item:1031}} as his items [also because people referenced these particular items on previous threads, wondering why anyone should get armor pen against "so little armor"] The first provides +20 armor, and the second +40, for an additional 60 armor. If you've paid attention so far, you should know this is already plenty for LDR to be effective, as it will remove the entire armor portion of {{item:3047}} and then some, even if Naut had no base armor; however, at level 9 he has an additional 64.3 armor, bring him to a total of 124.3 armor, of which he will effectively have only 80, after {{item:3036}} mitigates 43.5 armor. Even in this early game scenario, the Unique Passive of {{item:3036}} is worth an astounding 1,714.5g, making the whole item worth 3289.5g, or 117.5% cost effective. Again, {{item:3036}} becomes vastly *over*-efficient against a level 9 tank, with only {{item:3047}}, {{item:1031}}, and no armor from runes or other sources. Let's check again at mid game, say this Naut is level 12, and has his support core of {{item:3047}} + {{item:4302}} + {{item:3109}} + {{item:3050}} (his most common build from what I could find, and one usually completed between 15-20 minutes in, so seemingly a reasonable midgame point for a support tank) He will get 20 armor from {{item:3047}}, 40 armor from {{item:3109}} and 60 armor from {{item:3050}} for a total of 120 bonus armor, on top of his natural 76 armor, bring him to a total of 196 armor. Seemingly fairly tanky, eh? Well, not when you apply {{item:3036}}. It reduced that armor down by nearly 70, invalidating all the armor of {{item:3050}} + half the armor of {{item:3047}}, and leaving Naut with only 127 armor. This penetrated armor is worth 60% more than the previous chunk of armor lost, skyrocketting the value of {{item:3036}}'s unique passive, and the item as a whole as a result. Finally, let's look at a full build {{champion:111}} , and see just how little his armor is worth, and how much a single item destroys him. Most common end build for Support {{champion:111}} is as follows: {{item:3047}} {{item:3401}} {{item:3109}} {{item:3050}} {{item:3190}} {{item:3193}} I will do two calculations, one without the extra effect from {{item:3193}}, and one with him near three people, granting him the additional 40MR/armor from Stone Skin. Without Stone Skin we have 103 base armor + 20 + 40 + 60 + 30 + 40 = 293 armor. {{item:3036}} takes that down by over 100, to 190 armor, removing 2.4 times as much armor as in the first scenario, and making the Unique Passive ***MORE THAN PAY FOR THE ENTIRETY OF*** {{item:3036}} *before we factor in its 1575g worth of raw AD* With Stone Skin, {{champion:111}} gets an additional 40 armor, bringing him to 333, before its penetrated down to only 216, losing 115 armor in the process. ~ TL;DR: If the enemy team has any tank at all, even a support tank, get {{item:3036}} by your third item. If any one purchase any armor item at fucking all, even just a {{item:3047}}, guess what? That {{item:3036}} is more than cost effective, and will provide a larger damage boost than a second zeal item after you already have zeal + IE. TL;DR of the TL;DR Get {{item:3036}} as your third item every single game to optimize DPS. P.S. it used to be common knowledge, as well as common practice to get the old Last Whisper as the 2nd-3rd item for ADCs back when Last Whisper was literally identical to current {{item:3036}}, except with 5 less AD. It is *STILL* the best 3rd item for DPS, giving over a 20% total damage amplification against only 100 armor, and over 35% damage amp against 200 armor targets, *before factoring in its 45 AD*
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