Eye of Cthulhu is an easy boss, but not to a poorly geared player who is doing it for the first time. He can be considered a gear check of sorts, given you've got until morning to defeat him or he'll despawn. This is the first real boss fight of Terraria, and my guide's focus will be to suggest some gear and setups to help you through the battle and on to greater challenges. Eye of Cthulhu drops Crimtane or Demonite Ore, depending on which is present in your World, both of which can be used to make a better weapon, though defeat of the next boss - Eater of Worlds (Corrupt) or Brain of Cthulhu (Crimson) - will be necessary to make the Pickaxes and Armor Sets from these metals, as they require extra materials dropped by those Bosses.
Summoning & Random Fights
Eye of Cthulhu may attack randomly once you've hit 200 health and also have 10 Defense. You can delay this until you feel ready by simply not using the fifth Life Crystal that you find. Stay at 180 health while mining and searching through random treasure chests for some better gear. Once you've hit that mark, it can spawn any night, and is rather likely to (1:3 chance). You'll see the words, "You feel an evil presence watching you". At which point, it's good to go back to your base so you can fight it on flat land.
You can make a Suspicious Looking Eye to summon it manually simply by farming the demon eyes that attack at night time in Terraria. They drop lenses, and 6 of them will let you go to a Demon/Crimson Altar and bring up your crafting menu to make the Eye. Use it at night, and Eye of Cthulhu will attack immediately. These eyes can also be found randomly in chests throughout the world of Terraria. You can fight him as many times as you like, and even profit a bit by selling the ores once you've crafted all that you need.
Arena Preparation
You can do yourself a huge favor by simply making 3 long rows (a screen wide) of wood platforms, with a campfire in the center to provide its health regeneration buff. This gives you room to dodge its attacks and even make your own (especially if you're melee).
Boss Stats & Recommendations
Eye of Cthulhu has 2,800 life and 12 Defense. For someone unprepared, this can be a problem - I know firsthand. The first time I fought him when I was very new to Terraria, I nearly defeated him but morning came before he was dead. I simply had invested all my ores into pickaxes and good armor to protect me while mining. I hadn't taken the time to make a nice weapon. I recommend you use a ranged weapon for him, because it's twice as fast. A Gold/Platinum Bow with Frostburn or even Flaming Arrows is perfect - he's not immune to either of the debuffs those arrows inflict. Flaming Arrows are fine, and very easy to make. You can go Melee, but an arena is more important in that case - as is good armor and some healing potions - because you've got to take some hits to deal them and he likes to kite in a predictable pattern.
As for armor, a suit of Iron/Lead is the minimum I would use. The last time I fought him, wearing a full suit of Silver Armor, he was barely able to damage me due to the campfire's regeneration effects and the armor's damage reduction. Using Iron or Lead you can do just fine, so long as you have a good weapon and arena set up.
The Boss Fight
During the first phase of the fight with Eye of Cthulhu, he will attempt to fly through you three times, going quite far away when he is done making a pass. This presents a hurdle to melee players, and is why I recommend ranged. In the meantime, servants of Cthulhu will spawn - little flying eyes. These pose next to no challenge but may do a bit of damage before you can take them out. In all, they should be targeted down. They drop hearts on death, which will recover your life. Don't go out of your way to, but you can save them for the second phase of the fight when Cthulhu becomes more aggressive.
At Half Life
The second phase of the fight, Eye of Cthulhu switches to a more aggressive and demonic form and goes full melee. The servants stop spawning, if I recall. He will come at you more regularly, always three times. After his 3 attempts, you can freely attack him for a moment before he does it again. Having a cloud or blizzard in a bottle can let you double jump over his attacks, whenever you're able to get him coming at you horizontally. His defense drops to 0, so you can begin inflicting more damage with your attacks at that point. Learn the pattern, and the ore is yours. He's hardly worth farming unless you want a trophy/mask and only gives ores.
After the Fight
Eye of Cthulhu will no longer spawn randomly, though you can summon him again with the Suspicious Looking Eye. After defeating this boss, you are in a better position to take on the Eater of Worlds or Brain of Cthulhu depending on your world. You summon them by breaking three Crimson Hearts or Shadow Orbs with a hammer in the Crimson/Corruption. Every 3, the appropriate boss is summoned - so keep count. You need to down Eye of Cthulhu to do this, because Purification Powder bought from the Dryad NPC (or a lot of bombs) are necessary to break through the corrupted stones to get at the hearts/orbs. Defeating Eye of Cthulu will let the Dryad move in, so make a house for her. If you'd rather be a bit better geared before you take on those bosses, here's a tip: Head to the Corruption/Crimson and break just two with a hammer. You'll allow your world to randomly spawn Meteors, which drop somewhere in your world and provide metal for a good suit of armor and some weapons.
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