I run regularly for about 6 KM. I have completed multiple 10 K runs. I am targeting a half marathon this September. But when I run distances like 10 K + i get pain in my left calf muscle. I believe I need to strengthen my calf muscle. Please suggest exercises to strengthen the calf muscle.
First and foremost, you need to be sure that this pain isn't caused by some underlying problem that could result in injury. Does it only occur when you do a specific run at a specific pace etc? Does it feel like a 'pull' pain or just an overuse pain/ache? If in doubt, see a physio and get professional advice.
If you are sure it's just a general overuse pain then try the below as well as strengthening exercises (which I am sure someone more knowledgeable in this area will be able to provide you with details of). I used to suffer badly with overuse pain in my calves. These couple of things made a huge difference for me and I no longer suffer excessively with this.
Compression Socks or Calf Guards
Try them. They help stimulate blood flow in your muscles thus (in theory) removing by-products of exercise away from the muscle more efficiently resulting in less discomfort in the muscle. I found these really helped for me and I almost never run without them now. There are lots of brands out there to choose from. I use both these compression socks and these calf guards and find them both very good value and good quality.
The cheaper alternative to sports massage. Using a foam roller 3 times a week or so will give a similar effect to a sports massage and will help the muscle recover from running quicker than just resting. Have a read of this article for more detailed information about the foam roller and what it does.
Hopefully this has given you some other things to consider alongside strengthening the muscles. I know how frustrating this kind of pain can be!
Barefoot running is very good calf muscle training. Indoors is probably the best idea, outdoors i would suggest to run on soft surface if possible. There are specific barefoot style minimalist shoes but you don't need shoes if you have the right environment. Keep the distances low when you start cause the intensity is rather high.
As @Jimsan said compression calf sleeves are great way to help to ease the pain quicker than leaving it to settle on its own.
I often have pain in my right calf muscle and both techniques as Jimsan points out help. My calf muscle pain comes from my stride technique and only if I fore-foot strike during my stride.
I have tried to change my technique to attenuate the problem and land more mid-foot and it works. For me it's easier to land mid-foot on long runs, but with speed workouts and tempo runs, I'm usually on my toes... and the pain is back.
If you need to strengthen your calf muscles, perhaps you should consider strength training. All the other ways of addressing this (foam rollers, socks, barefoot, etc) don't really seem to answer your original question.
The most common (and I believe effective) lift is the obviously named calf raise. You can do this with or without a barbell on your back, dumbbells in your hands, or whatever other weights you choose.
The soleus and calf (gastrocnemius) are also used as stabilizers in the squat, and you'd be hard pressed to find a more effective lower body exercise. Rather than doing spot-work like the calf raise, I'd stick with a squat where you'll also strengthen your hips, lower back, abs, and complete leg in a balanced way.