The crescent kick is not flashy so may overlooked it as being just used as an exercise to loosen then hips. However, it has specific uses that make a useful part of your arsenal.
Outside Crescent Kick
Used mostly as defensive kick to block a punch, knife attack, or knock a gun away, or to open the guard for an attack. It is too easy to block to be used effectively as an offensive kick. Also used as a setup kick for a round kick with other leg. Kicker must be fairly close to opponent to use the kick. The higher the target, the closer you must be.
Foot position. Keep foot vertical, with toes upward.
Striking Surface. Strike with outside area of the foot, ankle, or leg
Leg Position. Leg may be kept straight throughout the kick or may be used in a snapping motion.
Movement. The kicking foot first travels inward and then it travels outward. The movement begins as a circular movement traveling inward and upward. At the height of the intended target, the foot travels horizontally and outward through the target. Then the foot completes its circular and downward movement back to the floor.
Motions. With the straight leg version, most all of the power comes from hip rotation. The kicking hip rotates inward, the straight kicking leg is raised upward toward the inside in a circle toward the outside, and the hip rotates outward pulling the kicking leg and foot through the target. The kicking foot moves in a large circle. With the snapping version, most all of the power still comes from hip rotation but a snap kick adds more power. The kicking hip rotates inward, the kicking knee is raised upward toward the inside in a circle toward the outside, and the hip rotates outward pulling the kicking leg and foot through the target as the leg snaps the foot into the target. The kicking foot moves in a narrow oval.
A jump, spin, or jump-spin may be added to the outside crescent kick.
Inside Crescent Kick
Same as outside crescent kick except that the leg and foot move from the outside toward the inside of the body, and the inside edge of the foot, ankle, or leg make initial contact.