Why did Paul shave his head at Cenchrea and not at Jerusalem as was the custom?

Answer:

"Paul, having remained many days longer, took leave of the brethren and put out to sea for Syria, and with him were Priscilla and Aquila. In Cenchrea he had his hair cut, for he was keeping a vow" (Acts 18:18).

Paul had his hair cut because he was keeping a vow. Luke doesn't mention what kind of vow, but it is assumed to be a Nazarite vow because that is the only vow that involves the shaving of the head. Traditionally the head is shaved at the start of the vow and then again at the end of the vow. The hair that grew during the vow was then offered as a peace offering to God. It is only at the end of the vow that hair is cut at the tabernacle and later at the temple. "Now this is the law of the Nazirite when the days of his separation are fulfilled, he shall bring the offering to the doorway of the tent of meeting. ... The Nazirite shall then shave his dedicated head of hair at the doorway of the tent of meeting, and take the dedicated hair of his head and put it on the fire which is under the sacrifice of peace offerings" (Numbers 6:13,18).

The rules in Number 6 do not say the hair must be cut at the start of the vow, but it is implied because if the vow is broken, the head must be shaved and sacrifices offered to restart the vow (Numbers 6:9ff).