From Sandi Marshall,
Your Guide to Crochet.
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In response to email requests for instructions on how to crochet a basic half circle, I've put together these directions and made example photos. This page shows a half circle in single crochet and the next page contains a pattern and photo for a half circle in double crochet.

Use thread size or yarn weight of your choice (with hook size to correspond). There's a pattern for 9 rows below. Stop whenever you have the size half circle you are looking for.

To start: ch (chain) 2, make 3 sc (single crochet) in the 2nd chain from the hook.
Row 2: ch 1, turn, 2 sc in ea of the 3 sc. (6 sc in this row)
Row 3: ch 1, turn, sc in first sc, 2 sc in next sc, (sc in next sc, 2 sc in next sc) twice. (9 sc)
Row 4: ch 1, turn, sc in ea first 2 sc, 2 sc in next sc, (sc in ea of next 2 sc, 2 sc in next sc) twice. (12 sc)
Row 5: ch 1, turn, sc in ea first 3 sc, 2 sc in next sc, (sc in ea of next 3 sc, 2 sc in next sc) twice. (15 sc)
Row 6: ch 1, turn, sc in ea first 4 sc, 2 sc in next sc, (sc in ea of next 4 sc, 2 sc in next sc) twice. (18 sc)
Row 7: ch 1, turn, sc in ea first 5 sc, 2 sc in next sc, (sc in ea of next 5 sc, 2 sc in next sc) twice. (21 sc)
Row 8: ch 1, turn, sc in ea first 6 sc, 2 sc in next sc, (sc in ea of next 6 sc, 2 sc in next sc) twice. (24 sc)
Row 9: ch 1, turn, sc in ea first 7 sc, 2 sc in next sc, (sc in ea of next 7 sc, 2 sc in next sc) twice. (27 sc)
To crochet more rows: On each new row, continue to add one more sc in between each increase (an increase is: 2 sc in next sc).