One way to strengthen the knees is to practice activating them in a variety of different positions.
A simple knee strengthening exercise that I sometimes use as a warm up in my yoga classes is to lift the heels while standing and then focus on making the knees feel strong. You could also think of this as “squeezing” the knees.
Try to make them feel strong at both the back and the front. Do this with the knees straight.
This same activation can be done while doing a standing forward bend.
With feet flat on the floor straighten your knees and make them feel strong at the front and the back. You can place your hands on the floor (or if you can't reach the floor, on chair or yoga blocks) or you can let your arms hand or reach them up behind you.
For a slightly different feet try it with the heels lifted. Have your hands on the floor for balance or try pressing evenly through toes and forefeet for balancing without using your hands.
Another knee strengthening exercise while bending forwards is to lift the fronts of your feet so that you are on your heels. Here again make your knees straight (or straighter) after lifting your forefeet, and then make your knees feel strong.
Straightening the knees and then making them feel “strong” or “active” can be two different and distinct steps. Straightening the knees doesn’t necessarily make them feel strong. Likewise, you don’t have to have the knees straight in order to activate them or make them feel strong.
Lifting the fronts of the feet while in a forward bend can be used as a calf stretch but it can also be helpful for strengthening the knee since one of the calf muscles, the gastrocnemius, crosses the back of the knee.
Lifting the front of the foot adds tension to this muscle by lengthening it and that can add tension to the back of the knee.
Likewise, lifting the heels adds tension to the back of the knee but it does so by causing the calf muscle to activate.
Chair pose can also be used as a knee strengthening exercise. You can work towards doing it with your thighs level.
Start with your thighs at about 45 degrees. Your knees will automatically feel strong or active in this position but experiment with making them feel stronger.
Alternatively, work at making the fronts of your knees feel strong. Then work at making the back of your knees feel strong (or in the case of this yoga pose, stronger.)
Again, because the calf affects the back of the knee you can also try squeezing the calf while doing chair pose.
I’d suggest do it at the same time as you are activating your hamstrings.
Two poses that are related because in them the back of the knee is closed is the deep squat and kneeling.
Both can be used as knee strengthening exercises.
If you have trouble with kneeling or squatting I’d suggest working on kneeling first since it’s easier to control how much pressure you put on your knees. In either pose practice activating your knees and relaxing them.
This page, yoga ankle stretches, gives some tips on how to work towards kneeling comfortably.
You may find these kneeling quadriceps stretches also helpful.
The video below also includes a helpful tip for strengthening the knee while doing an exercise called "The Seated Get Up". At about the 5 minute mark is a muscle activation that I've found very helpful in making my knee stable, even in a pose like a one legged squat (which is basically what the seated get up is.)
Once you can kneel comfortably, work towards a deep squat. If you have trouble do it near a wall or door or pole so that you can hold on or use the support to help balance.
To make it easier to work towards squat, first find a foot position and pelvis position which allows your hips and knees and feet to be comfortable. Once you find that foot position (wider, closer together, turned out to varying degrees or parallel) practice activating and relaxing your knees. Also try activating the front of your knees and then your back (individually.)
As you practice the deep squat, work at expanding your foot position possibilities. You may have to explore hip muscle activation at the same time. But in either case work towards the ability to squat with feet closer and perhaps even with feet parallel.
Other variations of the deep squat include shifting your weight to one foot.
Here again you can practice strengthening the knee by focusing on the leg that is bearing your weight.
I’d suggest shift weight first. Do it slowly and gradually so that you can stop if you feel any discomfort or pain.
With weight on your target foot, then squeeze the knee. Try activating the front of the knee, then try activating the back. You can also try activating both together. Relax and then return to center or repeat a few times and then work on the other leg.
Standing on one leg balance poses can also be used as knee strengthening exercises.Half Moon Pose
Half moon is one of my favorites. With your hand on the floor try to make the knee feel strong. Then shift your weight to your foot, make your foot feel strong and lift your hand.Single Leg Forward Bend
Another balancing on one leg pose for knee strengthening is the single leg forward bend.
Start with both feet and your hands on the floor. Shift your hips to the right so that your weight is over your right foot and lift your left foot.
Rather than reaching it back keep it hanging down. Shift your weight back. Press your forefoot into the floor, make the knee feel strong (the standing leg knee) and then lift your hands.
Another knee strengthening exericse that exercises the knees in a slightly different configuration is standing side bend.
With feet hip width push your hips to the left and bend your spine to the right. Optionally reach your hands past your ears to the right (or up and to the right.)
Shift your body so that weight is even on both feet. Then squeeze your knees. Because the hips are pushed to the side and the legs are angled, you may find it helpful to activate one knee at a time, the inside knee, then the outside knee.
Practice the knees individually, then practice activating them together.
Another standing knee strengthening exercise is standing twist.
With feet hip width and parallel turn your pelvis to the right, then turn your ribcage relative to your pelvis. Hold this position and and practice activating (and then relaxing) your knees. Here again you can try one leg at a time initially. Then try both knees simultaneously.
Two more of my favorite yoga poses that can be used as the framework for knee strengthening exercises are front triangle and triangle twist.
Both use the same foot position, one foot forwards the other back, with the back foot turned slightly out.
With hips square to the front foot bent forwards. Activate the front knee. Make it feel strong. For balance make your front foot feel strong. Then if you like lift your hands.
For triangle twist keep the foot position from front triangle.
With your right foot forwards place your left hand on the floor, lift your right hand and twist to the right. For this one you can keep your hips level, or push the hips back, or push them forwards. Activate and then release your front knee or if you like both knees. Do it with your breath. This can apply to all the exercises. Inhale as you engage the knee (or knees) and exhale as you relax the knees.
One of the main muscles that works directly on the knees is the quadriceps. The Conscious Muscle Control video:Quads (and superficial hip flexors) teaches you how to activate and relax the quadriceps (and superficial hip flexors) at will. You'll also learn to better feel your knees.
If you can already feel the tension in the front of your knees that active quadriceps creates, then you don't need this video.
Opposing the muscles at the back of the knee joint is the hamstrings. The Conscious Muscle Control Video: Hamstrings and Glutes teaches you how to feel and control your hamstrings and glutes. Both can have an affect on your knees.
One other set of muscles that works on the knees are the calf muscles, both the gastrocnemius and the soleus muscle. As part of your knee strengthening regime, try activating your calves in standing poses to see if calf activation makes your knees feel stronger or less subject to pain or both. Activate your calves to create a downward pull on your fibula (the smaller of the two lower leg bones), and to create even tension in the two hamstring tendons at the back of the knees.
As a yoga teacher, I'm constantly exploring new exercises, new ways of doing yoga poses.
There is no single "right way" of doing a yoga pose. Instead, there are options. And the better you are at "feeling" your body, the better you can get at choosing the right option for your body as it is now.
For any technique, the point of practice is to learn feel it and to control it, so that it can be used without thinking about how to use it.
And that is more or less the approach taken in all of my ebooks and videos. They help you to feel your body and control it so that you can work towards using it effectively in anything that you do.