"Half" or "A Half"

Solution 1:

Both are correct, with "one half" probably predominating in formal writing, and "half" predominating in speech and informal writing. Why this is correct is not precisely clear to me, except that informal speech often does not follow the same "rules" as formal speech.

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In speech, one often does not use "one/a" before half:

I'll be there in half an hour.
I don't know half these people.

Solution 2:

Normally a shortened form is used: half a pizza, half a loaf of bread, half an hour. One may assume that the full formula was "a/the half of a pizza". Even "The clock struck the half-hour" is used.