Because a major part of the ballet repertoire takes place on your toes, the ankles need to be strong to support your body weight. One exercise that you can do anywhere is ankle circles. Sit down, and cross one leg over the other. Circle the ankle of your top leg 20 times in each direction, and then switch legs. You can also write the alphabet in the air with each foot.
Another ankle exercise is springing from one foot to the other. Start with your feet together and your knees bent. Roll up onto the toes of one foot, and down onto the other foot. Settle on to the ball of the foot first and slowly roll down to a flat foot. The action of rolling through your feet will help strengthen muscles in your feet and calves as well.
Building strength in the calf muscles is inevitable in ballet because of the repeated action of rising onto your toes. To do additional strengthening in your calves, continue repeating the releve action, or rise to the toe action in various position. First, stand in parallel position or with the feet together. Next move to first position, with the heels together and toes pointed out at a comfortable angel. Repeat the calf raises in first position. Try balancing on one foot and doing the raises on each foot separately. This will help with strength and balance.
The plie position in ballet implies an outward bending of the knees while keeping the back straight. A grande plie implies a deep bend, bending the knees as far as you can go without raising the heels off the ground. Repeated grand plies will strengthen all of the major muscle groups in your legs. To take it one step further, stand in a wide second position, with your feet wider than shoulder width apart and toes pointed out. Lower into grande plie and spring back onto one leg with the other extended straight in front of you. Repeat this on both sides; it will help with strength and balance.
The arabesque and attitude positions in ballet require you to lift your leg behind you. The action of raising your leg behind your body, either in a straight or bent position, will help strengthen both your hamstrings and glutes. Start in a comfortable first position, with your heels together and toes slightly apart. Keep your leg slightly turned out from the hip as you raise your leg behind you. Repeat the raising acting on both legs in the bent and straight positions.