脸 (traditional 臉) is flesh 月 plus the sound 佥. Interestingly it's a latecomer — classical Chinese said 面 for face; 脸 originally meant just the cheeks, then took over the whole face in the spoken language.
脸 is anatomy AND sociology: 洗脸 (wash your face) starts the day, but 面子/脸面 (face — honor, dignity) runs Chinese social life. 丢脸 (lose face) is disgrace; 给面子 (give face) is showing respect by cooperating. 不要脸! ('doesn't want face!') is among the harshest insults — to be beyond shame. The physical face washes clean every morning; the social one takes a lifetime to keep.
The flesh 月 you present to the world — your FACE, in both senses.
丢脸 (lose face) entered English FROM Chinese — one of the few Chinese idioms to colonize English business-speak.