Will 12 inch subwoofer be enough for an 18x20 room (with 10 ft cielings)

I have a similar room; using JBL pro stage 18" powered sub, located on the back wall, firing sideways.  No complaints.

My living room is an open floor plan like yours with a 2 story cathedral ceiling. I have an older SVS 2039pc+ (12” driver with 500w amp) that produces plenty of earth shattering low bass. I use a Dspeaker Antimode 8033c to tame the peaks and take away room nodes. The improvement is very noticeable with cleaner multi-note bass. Instead of multiple subs get one very good sub and the 8033c.

http://www.dspeaker.com/en/products/anti-mode-8033.shtml

+1 ebanksms
When I compared a JL Audio F113 to my Velodyne DD-18 it was every bit as potent. At the time the newer JL seemed to have a better cabinet along with a proprietary driver. It's auto only ARO room correction was limiting. 

If you're able to crawl test and position a sub of similar quality and equalization you can expect very good performance at the listening position from a 10".
With this level of subwoofer adding a non equalized extension sub interconnected to the main sub and positioned at the rooms second largest mode will further reduce the rooms modes and maintain equalization.  

I am lucky.  I have a dedicated room.  I was very impressed by room treatments improving my bass.  I have 2-10" subs.  I don't think size matters as much as being able to have multiples and treatment.  It helps every aspect of the music.  I know, this could be a moot point in your case.  So if you walk around the room and hear peaks and valleys in bass and you cant do at least one (multiple or treat) try moving sub around. 

Depends on what you listen to and how loud you listen to it.

I have a 10 x 14 x 9 room and a 4 sub dba made up of three different brands; 2-13" subs and 2-12" subs. I don't listen real loud very often but on occasion I have pushed the room to 105 db plus and in doing so I have driven one of the subs, a Rythmik F12, into thermo overload twice. Is there a problem with the Rythmik you might ask? Maybe, but Rythmik did't think so. Am I a bass head/hog? When the music calls for it; absolutely. Am I completely crazy? No question, but the moral of the story is, It depends. Know your tendencies and what you are looking for and remember, the bigger the room the harder it is to fill.

I agree that sub location is critical, but only if you have options. If you can only put the sub in one place, that is where it has to go. I have one tucked under an end table and another behind the sofa. Are these the best locations for these subs? I have no idea, these were the only places I could put them, but the room does sound better with them than without them.