Hitting a punching bag seems like the most obvious and ideal way to measure and increase your power, but it's just one of many methods you have at your disposal. In truth, you can increase the speed and power of your punches using your mind and body as the only tools. By increasing overall strength and speed and improving your striking technique, you'll be punching harder in no time.
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Begin and end every boxing workout with a few rounds of shadow boxing. You can have all the raw power and athleticism in the world, but it won't mean a thing if you can't transfer that kinetic energy into an effective punch. Boxing isn't called the "Sweet Science" for nothing. Punches require full-body coordination, timing and fast-twitch muscle fiber.
Develop power and strength throughout your body by performing body weight exercises such as squats, pushups, crunches and planks. There isn't a single muscle most important in the art of delivering a powerful punch. Your entire body must act as a chain. Any weak link will affect the power of your strikes.
Work plyometric training into your conditioning routine. Powerful punches combine technique with speed and explosive strength. Box jumps, squat thrusts, long jumps and medicine ball slams create deep, fast muscle contractions that increase speed and power significantly.
Try shadow boxing while holding light free weights or wearing heavy boxing gloves. Adding a bit of resistance to your punching combos forces your muscles to work harder. That way, when you start punching for real, you'll be that much faster and more devastating when you make contact.