Odd math. I get 3,000 gp.
It would work just like the spell, so if you just spend your round looking at a person and deliberately detecting Evil, but then next round you don't, you only get 1 round of info. If you want the full info, you still have to spend your three rounds "using" the item.
For continuous item where you want the item to simply give the detect evil alignment/strength information after 3 rounds automatically of all eligible creatures in the area of effect, without needing to spend a standard action to concentrate....hmmm I'd say base price would be as you calculated (well your product is incorrect despite correct formula) 1x1x2000x1.5 = 3000 gp.
How would you work directional facing of the 60 ft cone if it's continuous? Would you have to aim it each round like the spell? Or is it gonna be 60 ft radius, centered on item?
If you gotta aim it, no modification on price. If 60 ft radius, I'd say multiply price by 4 as if using 4 detect evils aimed in 4 different directions simultaneously. Or would that be: 3000 + 3000x1.5 + 3000x1.5 + 3000x1.5 = 16500 gp due to 2nd, 3rd, and 4th directions counting as extra spell effects for extra 50% in cost.
However I think it would be annoying for DM to give out alignment of creature if it's evil or not simply being 3 rounds in character's presence automatically without character investing action on it.
Probably easier using use-activated instead of continuous when dealing with concentration spell-effects. Spell would normally require a standard action to maintain it each round, item probably should too. 1X1X2000 = 2000 gp.
The difference between use-activated and continuous isn't about whether it takes your actions. A use-activated item literally activates its powers when you use it. If it's a musical instrument, for example, then you have to play the instrument to get the effect, not merely carry it around. If it's a ring, you put the ring on to trigger the effect (although I've heard of "spinning" the ring around on your finger to trigger the effect). Etc.
Continuous items just work by having them equipped in whatever slot they occupy.
If you make this continuous, it means that the user doesn't have to use any actions to activate the effect. If you make it use-activated, then he must spend actions to activate the effect. Either way, the effect of 1, 2, or 3 rounds of ongoing detection to get more information would still require him to concentrate for 1, 2, or 3 rounds to get the appropriate info.
Let's say you make this as a magical ring. If it is continuous, then the user would simply need to wear it in the presence of someone whose alignment he wants to detect for three rounds, using his standard action to concentrate for the increased info each round. If it is use-activated, then user would need to wear it in the presence of someone whose alignment he wants to detect for three rounds, using his move action each round to take it off and put it back on (or maybe to spit in around his finger, or something to physically count as "use" the ring) and also using his standard action to concentrate for the increased info each round.
Note, this means that as a continuous item, someone could still move around town, looking for evil, though he'd be moving at half speed since he only gets one move/round instead of two. As a use-activated item, moving around town is a bit harder since he'll only be able to take his free 5-foot step every round.
This extra action to "use" a use-activated item is why it's priced cheaper (except for very long duration spells in which case it's priced the same).
As for cone vs. radius, I can't think of any reason to turn it into a radius because that's not how the spell works. It's not so much about facing, but rather about the area the magic can involve with each action spent "studying" the are. Or to put it another way, turning this cone into a radius would be (almost) as absurd as turning a red dragon's fiery breath into a radius spreading out from the dragon in every direction. I suppose you could argue for it, and even justify it, but clearly that would make the dragon more powerful and it would make the item of Detect Evil more powerful (and presumably more expensive) - but there are no RAW rules for this that I know of, so it's GM judgment.
Some outsiders have "Constant detect [alignment]". It provides first-round information automatically, and the option to concentrate for second- and third-round info.
Call it a rounds-per-level effect - that's how you'd usually use detect evil anyway - for a cost of 4*2,000*1*1. Seems steep, though. Gut feeling tells me 5k is more like it.
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A Shining Wayfinder costs 1 500 gp more than a standard Wayfinder.
In addition to what a standard Wayfinder does, it has the ability to Detect Evil (but only in the hands of a Good creature) and can cast Protection from Evil once per day.
I do not know how the formula works, but I guess this can provide a solid basis to estimate the value of a Detect Evil item.
Shining way finder sounds exactly like what OP wants. Just when you have it out spend a standard action each round to read it as if using a regular wayfinder to avoid becoming lost.