Learn Hardcore Korean: mian, giri makhyeoseo sip bun jeongdo neujeul geoya. - Sorry, the traffic is heavy, so I’ll be about ten minutes late.

미안 is a clipped, very casual apology, common in texts among close friends.
미안해 is still casual but a full sentence; safe with friends/peers.
미안해요 is polite and friendly; use with acquaintances or anyone you should show some respect to.
죄송해요 / 죄송합니다 are more formal/deferential; use with seniors, strangers, or in professional situations. In this sentence, 미안 signals casual texting style. To be polite, you could say: 미안해요, 길이 막혀서 10분 정도 늦을 거예요. or more deferential: 죄송해요, 길이 막혀서 10분 정도 늦을 것 같아요.

Why is it 길이 막혀서? Can I say something else like “traffic is heavy”?

Korean commonly says:

길이 막히다 (the road is jammed)
차가 막히다 (cars are stuck)
교통이 막히다 or 교통 체증이 심하다 (traffic is heavy/severe) All are natural. Your line uses 길이 막혀서, which is very idiomatic. Synonyms you might see in messages:
차가 많이 막혀서…
교통이 너무 막혀서…
차가 많이 밀려서… (cars are backed up)

What does the ending -아서/어서 in 막혀서 do here?

It connects a reason/cause to a result: “because/since the road is jammed.” So 막혀서 늦을 거야 = “Because it’s jammed, I’ll be late.” Compared to -니까:

-아서/어서 gives neutral background reason; very common in explanations.
-니까 can sound more direct/justificatory and is common before suggestions/commands. Both can work, but -아서/어서 feels smoother here.

Is 막히다 passive? Who’s doing the action?

Yes. 막히다 is the passive counterpart of 막다 (“to block”). With traffic, Korean describes the state as “being blocked/jammed” rather than “someone blocks it.” So 길이 막히다 = “the road is jammed.”

Do I need the subject marker 이 in 길이? Can I drop it?

Full form: 길이 막혀서…
In casual speech, particles are often dropped: 길 막혀서… is common in texting.
You could also use 길은 막혀서… with 은/는 to contrast or topicalize (“as for the road, it’s jammed…”).

Should it be 십 분 or 열 분 for “10 minutes”?

Minutes use Sino‑Korean numbers, so say 십 분 (10 minutes). 열 분 is usually understood as “ten people” with the honorific counter 분 for persons (한 분, 두 분, …). Avoid 열 분 for minutes.

What does 정도 add in 십 분 정도? How is it different from 쯤?

Both mean “about/approximately.”

정도 is a dependent noun placed after the amount: 십 분 정도.
쯤 is a particle that attaches to the preceding phrase: 십 분쯤. Nuance is minimal: 정도 can feel a bit more neutral/formal; 쯤 is very common in speech. Use one approximator at a time. Alternatives: 약 10분, 한 10분 (colloquial), 10분 가량, 10분 내외 (slightly formal).

How do I space and write numbers here? 십 분, 10분, 분쯤, 정도?

Spelled‑out numbers + counters: add a space → 십 분.
Digits + counters: no space → 10분.
정도 is a separate word: 십 분 정도 / 10분 정도.
쯤 attaches directly: 십 분쯤 / 10분쯤 (no space before 쯤).

Does “10 minutes late” mean “I’ll arrive in 10 minutes” or “I’ll be 10 minutes later than planned”?

It means “I’ll be late by about 10 minutes” (relative to the expected time). If you want “I’ll arrive in 10 minutes,” say:

10분 후에 도착할 거야.
10분 안에 갈게. (I’ll be there within 10 min.)

What exactly is -(으)ㄹ 거야?

It’s the periphrastic future/probability form: “will/is going to.” It’s built from -(으)ㄹ 것 + 이다 → -(으)ㄹ 거야 (casual). Polite equivalents:

-(으)ㄹ 거예요 (standard polite; note the spelling 거예요, not “거에요”)
-(으)ㄹ 겁니다 (formal)

How would I say this more politely or with softer tone?

Polite:

죄송해요, 길이 막혀서 10분 정도 늦을 것 같아요.
미안해요, 길이 막혀서 10분 정도 늦을 거예요. Softer/hedged:
Add 좀/조금: 조금 늦을 것 같아요.
Add 아마 (probably): 아마 10분 정도 늦을 것 같아요.

Difference between 늦을 거야, 늦겠어, and 늦을 것 같아?

늦을 거야: straightforward future/prediction (“I’ll be late”).
늦겠어: speaker’s immediate judgement/speculation or exclamation (“Looks like I’ll be late,” “I’m going to be late!”).
늦을 것 같아: hedged/sounded‑out guess (“I think I’ll be late”). Polite: …같아요.

Is 막혀서 the only natural connector? What about 막혀 가지고?

-아/어 가지고 is very common in colloquial speech to indicate cause/result:

길이 막혀 가지고 10분 정도 늦을 거야. It’s informal; in writing or more formal speech, -아서/어서 is preferred.

Any pronunciation tips for this sentence?

십 분 undergoes tensification: pronounced roughly [십뿐] (the ㅂ of 분 becomes strong).
막혀서 is [마켜서].
거야 is [거야]. Natural rhythm groups it as: 미안 | 길이 막혀서 | 십 분 정도 | 늦을 거야.

Can you break down the sentence parts?

미안: “sorry” (casual interjection).
길이: “the road” + subject marker 이.
막혀서: “is jammed” (막히다) + -어서 (“so/because”).
십 분: “ten minutes” (Sino number + minute counter).
정도: “about/approximately.”
늦을: “will be late” (future adnominal of 늦다).
거야: contraction of 것이야 (copula), forming the periphrastic future -(으)ㄹ 거야.