Stair Math

My family has been working on some stair math. My sister is climing the stairs in her office building. We want to know the distance she is going. Here is a summary of what we have decided - in case any visitors want to contribute. We'd love to hear some other way to measure.

When A is going up stairs the true distance she travels should be the diagonal distance - because she is going up AND across. Think of it this way: If you put a plank of wood on the stairs to make them flat, that is the distance she would have walked - like going up a hilly road. To get the measurement of that distance, you can calculate it knowing the height and depth of the stairs.

You have to assume that there is a right angle on the stairs where the height and depth meet - but that is a good assumption, or you would notice a tilt to the stairs. (We also assume that each stair has the same dimension, so once we know one stair, we can know the sum of the stairs.) Use Pythagoran Theorem and you get that the square root of the height squared plus the depth squared is the hypotenuse (the distance traveled). If it is 8 inches high and deep, that is ~11 inches per step, so with 11 flights of 24 steps each, you are at ~0.05 miles.

This is the correct way to measure it - not the up and the across, because she is not shuffling her feet. A - you do not have to measure the arc your foot makes. We don't do that when you are walking outside to get distance.

Unfortunately, as nice as the pedometer quick calculation was, it was WRONG. Sorry to get the hopes up. I told you the average stepping distance was 2,000 steps per mile. That would mean a step/stride of ~32 inches per step. As we stated, the hypotenuse is only ~11 inches. At 11 inches per stride, you have to go 5760 steps per mile. Since you only go 264, you are (again) at ~0.05 miles per your 11 flights of stairs.

Any other ideas on how to get a nicer number? Maybe we should look at heart rate, and results of blood pressure and lactic acid production? Going up those 0.05 miles of stairs is way harder than going 0.05 miles to the coffee machine.