Manamune and Archangel’s Staff are a couple of items that grant bonus attack damage (AD) and ability power (AP) as a percentage of the champion’s mana. A good question is if they are worth purchasing in terms of the pure bonus AD and AP that they provide. Let’s do some math and look for an answer.
The bonus AD and AP provided by Manamune and Archangel’s Staff can be calculated as a function of both the champion’s current mana pool: base mana plus mana from other items and the variable bonus mana given by Manamune and Archangel’s Staff. By itself, Manamune provides a minimum 350 mana up to a maximum bonus of 1350 mana; whereas, Archangel’s Staff grants a minimum of 400 mana and a maximum of 1400 bonus mana.
Because of the dependence on base mana, it becomes relevant to notice that, as listed by the League of Legends Wiki, Ashe and Sion are tied for the smallest mana pool, 200 mana, at level 1. Moreover, Janna has the biggest mana pool both at level 1 and level 18 with a size of 366 and 1454 mana points, respectively. Finally, by level 18, the smallest mana pool corresponds to Sion with 625 mana points.
If the current mana is CM and Manamune’s bonus mana is MmBM then the total bonus AD is MmAD = (CM + MmBM)*0.02 + 20. That is 2% of maximum mana plus a 20 flat bonus to AD.
On Archangel’s Staff’s case, if its bonus mana is ASBM then the bonus AP becomes ASAP = (CM + ASBM)*0.03 + 45; 3% of maximum mana plus 45 bonus AP.
Table 1 shows the bonus AD granted by Manamune in the possible range of base mana. Manamune’s bonus AD can be easily reached by performing attacks and using skills. In that case the bonus AD goes from 51 to 77, depending on current mana. Even considering that carries tend to have small mana pools, by level 18 Sion’s minimum of 625 mana points ensures a respectable 59.5 bonus AD.
Table 1:
Table 2 shows the bonus AP afforded by Archangel’s Staff in the plausible range of base mana. Even for a champion with as little mana as 200 Archangel’s Staff provides a decent 93 AP while helping to solve mana problems; at its full bonus mana. Regardless, there’s no doubt that champions with larger mana pools like Janna will find more value on Archangel’s Staff.
Table 2:
Graph 1 and Graph 2 show the behaviour of the bonus AD and AP as a function of the current mana and the bonus mana. In each case it’s plain to see that the increase is linear: more mana means more AD and AP, in each case.
Graph 1:
Graph 2:
To answer the question we posed at the beginning we can use Long Sword, Pickaxe and B. F. Sword as a basis to calculate the cost of AD: one AD costs 42, 39 and 37 gold; respectively. The same can be done for the cost of AP using Amplifying Tome, Blasting Wand and Needlessly Large Rod: the cost of one AP is 21.8, 21.5 and 20 respectively.
Manamune costs 2110 gold, therefore it should be worth between 50 and 57 AD; the latter being reachable with a CM of 500, considering full bonus mana. Archangel’s Staff costs 2855 gold, in consequence it should be worth between 131 AP and 143 AP. That means that full bonus mana plus a CM of 1500 is necessary to make Archangel’s Staff worthwhile in terms of pure AP.
In conclusion, Manamune is a cost effective item for building AD. Not only does it increase attack damage but it also helps solve mana problems at an affordable price. On the other hand, Archangel’s Staff, in terms of pure AP bonus, is not such a worthwhile item. It does help a lot with mana by increasing the pool and adding regeneration but it all comes at a steep price.