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In addition to cardio exercises, you can also include some strength training exercises using the stairs as well. Working your legs and glutes is particularly easy with stairs.  Lunges are an easy exercise to adapt to a staircase.  Lunges really work your legs and glutes without the addition of stairs, so doing them on stairs will really up the intensity.  To do lunges, you'll take the stairs two or three at a time.  Going for more stairs than this will increase your risk of overstraining. Step your right foot up two or three stairs.  Focus on pulling yourself up the stairs using your right leg.  You will notice this in your thigh muscles.  Push yourself up until your left leg comes to meet the right leg on the step. Repeat either on the same side or do alternating lunges.  Aim for 10 lunges per side or do as many as you're able. Running or jogging up the stairs is pretty taxing on your legs, glutes, heart, and lungs. If you want to get a balanced workout, you're going to want to include some upper body work as well. The elevation of the actual stairs also allows you to work the backs of the arms and triceps.  To start, face away from the staircase.  Rest your arms on the second or third stair with your fingers facing away from the stairs.  Grab the corner of the stair for this exercise and keep arms shoulder width apart. Keep your feet flat on the ground in front of you.  Press your hips up so that your arms are in a straight, extended position. Slowly lower your body down using your triceps to allow your body to dip down toward the stairs.  Lower down until your arms are bent about 90-degrees. Push yourself up back to the starting position. Try to do three sets of 15 to 20 reps. Like tricep dips, you can also use the stairs to do push-ups as well. These work your arms, chest and core muscles..  Start by facing the stairs and placing your hands on the first or second step.  Keep your legs extended behind you so you're almost in a plank-like position.  The higher the stair you use to place your hands, the easier this exercise is. While keeping your hands shoulder width apart, slowly bend your elbows so that your upper body and face slowly lower down towards the stairs. Lower down until your nose is almost touching the stairs.  Hold this position here for a few seconds and then slowly push yourself up to the starting position. Do a decline push-up variation by placing your feet on the bottom or second step and your hands on the floor at ground level. Similar to lunges, doing side steps up the staircase can help strengthen your legs, but more specifically the inner and outer muscles of your thighs.  Start this exercise by standing next to the stairs with your right shoulder facing the stairs. Step up 1 or 2 steps with your right leg and bring your left leg up to the step below it. Repeat until you get to the top and then do it again, this time leading with your left leg. This exercise can really make your calves pop. It works the gastrocnemius, which is likely what you think of when you imagine a calf muscle.  Start by standing on the edge of the stair. Only about the first fourth of each foot should be on the stair; your heels should be hanging off the edge. Rise up on your toes as high as you can. Keep your back, legs, and feet straight and don't allow yourself to lean forward or backwards. Slowly lower your body as far as possible. Keep one hand on the banister if balance is a concern.  Try doing this on one leg for an even more effective exercise, but only do so if you have a banister or wall to hold.
Try stair lunges. Do stair tricep dips. Use stairs to do push-ups. Try climbing side steps. Perform stair calf raises.