There are many exercises designed to increase tricep strength. Doing exercises that isolate the triceps are probably the best way to build strength quickly and in the long term. Tricep exercises can be done with free weights, cable machines, or no equipment at all. The triceps are a group of three muscles at the back of the arm, necessary for straightening the elbow. Thus, most triceps exercises focus on movements that extend the arm, or both arms at once.
To begin to increase tricep strength without any equipment, start with dips and narrow push-ups. Narrow push-ups are a slight modification of the push-up in which the hands are placed inside the width of the shoulders. Bend towards the floor until your chest touches your hands, then push back up. A more advanced version of the narrow push-up involves placing the thumbs and first fingers together to form a diamond on the floor, which works the triceps even more. As with almost all other tricep exercises, keeping the elbows as close to the body as possible will maximize the effect on the triceps.
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Dips can be done with a chair, with a pair of benches, or at the gym with a dip apparatus. To use a chair, stand with your back to the chair and place your hands behind you on the seat of the chair. Extend your legs out in front of you keeping your knees slightly bent, then lower yourself until your upper arms are parallel to the floor. Push back up and repeat. Using a pair of benches or dip apparatus is more difficult, but will increase tricep strength more.
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A bench press or a barbell give you more options. For one way to work the triceps, lie back on the bench press, grasp the bar with the hands shoulder-width apart, and lower the bar until it reaches the bottom of the pectoral muscles. Keep the elbows close to the body and forcefully push the bar back up. On a plain bench, use the same technique but lower the barbell to the forehead, then push back up. Similarly, you can use dumbbells on a bench. Begin with arms extended upwards, palms facing each other, then lower the dumbbells to forehead height before pushing back up.
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You can also increase tricep strength by using a cable machine. One method is the tricep press-down or push-down. First, set the cable at a high pulley. Begin with the forearms parallel to the floor, then push down until the arms are extended. Slowly raise the arms to the starting position. Using a reverse grip on the this exercise is easier, while using a rope attachment or one-handed push-downs are harder.
There are many more ways to increase tricep strength. There is a great variety of triceps exercises, particularly for barbells and dumbbells. You can join a gym or try weight training books to find the ones that suit you best.