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

I've used 2 x 18" Eminence Kilomax drivers is 14 cu ft sealed cabinets in a room of similar volume.  EQ'ed the subs and placed them in the corners behind the main speakers.  Works OK. 

It all depends on the listening position in the room.  The optimum listening position and main speaker position may not be the optimum listening position for the subs.   

The answer to your question is NO.  Get four subs...it will be for the best.

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.