Half hexagons are used to fill in the gaps at the edges of joined hexagons. Unlike their full sized counterparts that are worked in rounds, half hexies are worked in rows.
In my experience, half hexies are harder to get perfectly in shape, they require a bit of blocking or edging to pull that turning line straight. However, its easily done and in no time you have a perfectly-formed space filler.
Row 1: Starting with a magic loop, 3ch (counts as 1tr), [1tr, 2ch, 1tr] twice, 1tr. Turn. 6 sts.
Row 2: 3ch (counts as 1tr), [1tr in each tr to the 2chsp, (1tr, 2ch, 1tr) into the 2chsp] twice, tr to end. Turn. 10 sts.
Continue to work row 2 until you have the desired number of stitches.
Give the hex a good blocking.
You can join your hex to others using the technique here.