2010 Specialized Tricross BB size I have a 2010 Specialized Tricross Sport Triple. I want to know what type and size of Bottom Bracket I need to replace the original one. 

Specialized's archived page for this bike lists the BB as a "BB-7420 w/crank bolts". From what I've read online, this is a British threaded, square tapered, 68mm BB. 

From what I've read elsewhere online, a common replacement for this is the Shimano BB-UN55, at around $15 CAD online. However, I haven't found a consensus about what length the Tricross needs. Here someone says 118mm for the 2011 Tricross; Bikepedia lists 113mm for the 2010. 

Is there any way to figure that out without taking the bottom bracket out to read the specs off of it? 

And is there any other spec beyond the threading, width, taper style, and length, that I need to ensure compatibility? 

Finally, if there's a better replacement than the BB-UN55, I'd be curious to know about it. I'd like to stay under $50 CAD. 

Thanks.
The typical thing to do here would be pull the cranks and measure the spindle length with a caliper. You don't need to pull the BB itself, all you care about is the spindle length. (On older bikes you'd also care about whether it was symmetrical or not, but yours will be).

There are some tricks you can play if that is not an option.

If the spindle is hollow, you can remove the crank bolts and poke a wire, spoke, QR, or string all the way through, mark it, pull it out, and know your spindle length that way.

If it's not hollow, you can remove the crank bolts and then use a caliper or pocket ruler to measure the distance between the tip of the spindle and the outside surface of the crank. Then you could use a couple straight-legged chairs or similar pieces of furniture, or some big squares, pushed up against the surfaces of the crank on either side of the bike to measure the spindle. Sandwich the bike between them, gently lift it up, measure the distance between the two surfaces, subtract the inset distance of the spindle tip, and that's your spindle length.