Overview
There are few things in karate more aesthetically pleasing than a perfect head-height round kick delivered during kumite. Perhaps that's because there's no easy way to learn this kick - it depends on hard work and persistence. If you're not flexible enough to lift your leg very high, (and most of us aren't when we start out), there's no shortcut route to greater strength and flexibility. You have to stretch dilligently in class, and you have to push yourself a little outside your comfort zone every time you do so.
Time after time, I see people start their attacking combination with a round kick, only to be surprised when they are punched long before the kick lands. A round kick is a powerful weapon, but for all but the most skilled students, it's too slow to use as a first attack. It's much more effective in the middle of a combination, perhaps following a side backfist or punch.
Round kicks are useful because their circular approach path swings them in behind many people's guards. They're great for attacking ribs exposed by a guard too far from the body, and a round kick is great to attack a head that's left vulnerable by a low guard.
Used in these katas
None up to black belt
Technique
Start in parallel stance with your guard in front of you. Lift your right knee to the side until your upper leg is horizontal. Ensure that you hold your shin up behind your knee, parallel to the ground. A viewer from the front should not be able to see your foot because it should be hidden behind your knee. Now swing your knee in a 90 degree arc forwards, and as it reaches its destination, swing your foot out from behind your knee, all the time keeping your thing and shin high.
Although it's preferable to keep your supporting foot facing forwards, most people find it impossible, and have to rotate that foot and the hip, by as much as 180 degrees. The further you rotate your hips and foot, the higher you will be able to kick, but the more you compromise your position, and the more you delay your recovery.
Round kicks can be delivered with the top of the instep or with the ball of the foot. The ball of the foot is more powerful, but the instep is easier. In kumite, you are not allowed to kick head-height with the ball of your foot, as it's too powerful and dangerous. However, some judges will not score a body kick unless you use the ball of the foot. You are expected to use the ball of your foot when practicing in class.
The more horizontal your striking foot at the moment of impact, the less chance that it will be deflected upwards by a block.
Having struck your target, you should pull your foot back as quickly as possible to prevent it being grabbed. Some people say that it doesn't matter if you drop your foot back to the ground in front of you as you're unlikely to do two consecutive round kicks, but you should train yourself to pull your foot back to the side, if for no other reason than the fact that it improves balance.
Opinions vary on what to do with your arms as you deliver your kick. However, many experienced martial artists like to push the kick-side arm out in front of them (above the kicking leg), in order to aid stability, and to provide cover for groin or head if needed. At the same time, the other arm remains bent in a guard, with the elbow pressed tight to the opposite hip, where it provides a back guard. I find that holding my arms in a guard this way also helps me to keep my stomach tensed.
The alternate arm action suggests that you swing both arms in a 90 degree semi-circle as you kick, starting from the opposite side to your kicking leg, and coming around to meet your leg from the opposite direction. This has the effect of counter-balancing your leg, increasing your speed, and providing a smoother kicking action. However, it is very unwieldy, and tends to leave you exposed on that side of your body. After the kick, your arms should be swung back in the opposite direction until you reach a normal guard, which helps to retract your leg faster.
Common mistakes to avoid
The commonest mistake is simply not lifting the knee to the side. Students often get confused between this and a front kick, and as a result, they lift their knees to the front. It's a valid technique, but it's not the right technique.
Another similar mistake is to prepare the knee to the side properly, but then to drop it as you kick, turning your round kick into a front kick. This is a useful tournament dummy, but is not a proper round kick.
The round kicks gets its power by using the the two-stage swinging action of the leg - first the thigh, then the shin. If you straighten your knee too early, you lose power on the last stage of the kick.
Remember when you kick, not to flap your arms around. They will unbalance you and reduce your effectiveness.