火=很受欢迎，受欢迎 is little less popular than 火.

ATTENTION: 火 and 受欢迎 are NOT the same thing even if both of them mean "popular". 受欢迎 means someone or something becomes popular because it's very nice. If someone did something very good, he will become 受欢迎. For example, Martin Luther King, Jr 很受欢迎. ---- because he did something very good. If something is liked by public, it will become 受欢迎. For example, this film is very very good, it 很受欢迎. about 火: 1) you can replace all 受欢迎 above mentioned with 火. It's the same. 2) 火 can be used for other cases. for example, a girl put her nude photos on her facebook. So everyone knew her. This girl becomes 火 --- we say: 这个女孩火了. BUT it's not because she is nice or she did something good, it's because all people watched her nude photos --- this is how she became popular. In this case, you can just use 火. However, 火 can't be used to describe illegal things. For example, if someone killed 100 people so everyone has known him, we CAN'T say 他火了. so, 火 can be used to describe something good, something bad but NOT illegal. so we can say 火 >= 受欢迎. if something is 受欢迎, it is definitely 火. but if something is 火, it may be 受欢迎 or may be not.

for the most part you are right. When we say 火， it usually refers to someone or something is known by a lot of people, such as a celebrity. But it dosen't necessary 受欢迎. 受欢迎 could describe anyone who is liked by some people, like your teacher. There is an extent difference.

是的， as popular as fire! VERY HOT

“火”是受关注多，热门的意思。但不一定受欢迎。“受欢迎”有人们喜欢，赞赏的意思。