Greetings, forum-goers. This is a short guide on getting the most defense for your dollar. It's not particularly complicated, but I feel it's something that everyone should know. So, here it is.

Well, we have three stats in League of Legends that determine our characters' survivability. These are:

--Health
--Magic Resistance
--Armor

Health is pretty simple. The more health you have, the more punishment you can take.
Armor decreases the amount of damage you take from physical attacks.
Magic Resistance decreases the amount of damage you take from magical attacks.

Armor and Magic Resistance are useless without health. The converse is less true, but still pertinent. This is because Armor and MR make your health more effective. Each point of Armor or MR increases the amount of damage of that type (physical or magical) needed to kill you by 1%. So, if you have 500 HP, adding 1 armor will increase your effective health against physical attacks by 5. If you have 5000 HP, adding 1 armor will increase your effective physical health by 50.

So, if our arbitrary goal number is 7500 EHP,
5000 HP, 50 Ar/MR: 7500 EHP (5000 * (1 + 0.5))
3000 HP, 150 Ar/MR: 7500 EHP (3000 * (1 + 1.5)
2000 HP, 275 Ar/MR: 7500 EHP (2000 * (1 + 2.75)

One of these is cheaper than the others. I suspect that you can guess which one.

Health, Armor, and Magic Resistance each have two basic items. These are:
-- Ruby Crystal: 180 HP for 475 gold (2.64 gold per HP)
-- Giant's Belt: 430 HP for 1110 gold (2.53 gold per HP)
-- Cloth Armor: 18 Armor for 300 gold (16.67 gold per Armor)
-- Chain Vest: 45 Armor for 700 gold (15.56 gold per Armor)
-- Null-Magic Mantle: 24 Magic Resist for 400 gold (16.67 gold per MR)
-- Negatron Cloak: 48 Magic Resist for 740 gold (15.42 gold per MR)

For the sake of consistency, I'll use the average of the numbers above;
--HP costs 2.59 gold per point
--Armor costs 16.12 gold per point
--Magic Resist costs 16.05 gold per point

Armor and Magic Resist do *not* exhibit diminishing returns; their effect is based solely on your health.

When is it better to buy Armor and Magic Resist, and when is it better to buy health?

A few things to take into account:
--Does your champion have a defensive steroid of some kind?
Example: Rammus benefits more from health because of his Defensive Ball Curl.
--Does your champion have a healing factor or a shield of some kind?
(Shields are essentially just extra health with a temporary duration)
Example: Skarner benefits from high resistances due to his Seismic Bastion.

This is as good a place as any to mention that certain champions get extra benefit from building specific resistances. Shen and Vladimir get more damage when they build health. Malphite gets damage for building armor, and gets more health for building health. Galio gets fully half of his Magic Resistance as AP.

I recommend that you take these things into account when consulting the following information. What I'm giving you is the point at which about 6 points of HP will get us the same benefit as 1 point of Armor or MR (this is roughly the change in EHP that you can get for about 16 gold).

And here's some math for you math people:
EHP with 6 more HP vs. EHP with 1 more Armor/MR
(HP + 6) * (1 + (AR/100)) = HP * (1 + (AR + 1)/100)
This assuming we increase only one stat, however. If we want to increase both evenly, we double the HP interval.
(HP + 12) * (1 + (AR/100)) = HP * (1 + (AR + 1)/100)

And we get, for a given HP value,
Even Defenses (1 stat) = (HP - 600)/6 = 50 Ar or MR per 300 HP past 600
Even Defenses (Both stats) = (HP - 1200)/12 = 25 Ar + MR per 300 HP past 1200

HP: --> 1 stat --> Both Stats
600 --> 0 --> n/a
900 --> 50 --> n/a
1200 --> 100 --> 0
1500 --> 150 --> 25
1800 --> 200 --> 50
2100 --> 250 --> 75
2400 --> 300 --> 100
2700 --> 350 --> 125
3000 --> 400 --> 150
3300 --> 450 --> 175
3600 --> 500 --> 200

And so on. The pattern is fairly simple.
(Armor and MR have very similar prices; for the sake of space, I'm going to condense them)

Well, this is pretty simple. If you get 25 Armor and MR for every 300 HP by which you exceed 1200, you'll be in pretty good shape. If your HP doesn't exceed 1200, get more health.

This only applies, of course, if your opponents are dealing split damage. In a lane against an all-physical or all-magical enemy, Armor and MR become dramatically more useful starting at 600 HP. Be sure to take into account who is dealing the damage on the enemy team; 4000 Magic Resist isn't going to do you any good against a fed Talon.

If your opponent is building to reduce your effective stats, you may need to increase those stats to compensate.

This works in two ways:

--Flat Penetration: Sorcerer's Shoes, Abyssal Mask, Haunting Guise, The Brutalizer, Youmuu's Ghostblade, and those pesky Greater Mark of Magic Penetration/ Greater Mark of Desolation/ Greater Mark of Precision.

When an opponent has one of these, the only way for you to compensate is by adding more Defense. It means that your build will sub-optimal when fighting enemies without these stats, so gauge carefully whether the champion in question is particularly threatening to you.

--Percentage Penetration: Last Whisper and Void Staff are the biggies to watch out for here. Most champions will have at least one of the masteries that grant % penetration; there's nothing you can do about that. These nasty items decrease the efficiency of your Armor and Magic resist by about half; rendering thousands of gold worth of items nearly useless instantly. When an opponent has one of these items, if you're still dying in fights you need more health. It might even be worth trading out an Ar/MR item for more HP if the character in question is a serious threat.

To you, the reader, for perusing my early morning mathematical musings. Send me an upvote if you find this guide useful, and leave questions or comments if you have any.

Oh, and one more word to the wise: more is better. 1200 HP and 0 defenses is by no means the best strategy; it just means that you're even on HP and defenses.