Q:

Pronunciation of 日 in compound words

I learnt from my teacher that the days of the month from 1 to 10 use kun and irregular pronunciations:
一日: ついたち (irregular)
二日: ふつか (kun)
三日: みっか (kun)
...
十日: とおか (kun)
But for words like 一日目, 二日間 and 三日間後, what would be the correct pronunciation and the rules for this?
I heard 一日目 is いちにちめ and 三日間後 is みっかかんあと. Is this correct?

A:

Yes, what you've learned is correct.  The first ten days "always" (that is, I've never found otherwise) use the kun readings, except for 一日 as you noted with 一日目.  Also, for "later/after", the pronunciation is ご, not あと.

今日は誕生日の四日前（よっかまえ）です。　→　Today is 4 days before my birthday.
九日間（ここのかかん）かかった台風　→　A typhoon that lasted for 9 days
クリスマスの六日後（むいかご）は大【おお】晦【みそ】日【か】です。　→　6 days after Christmas is New Year's Eve (Ōmisoka). / The 6th day after Christmas is New Year's Eve.

