Question

How do I convert 35.2 ft^2 to cm^2 using dimensional analysis? ie, what are the steps when you have units squared?

Answer 1

1in = 2.54cm
so, 1in^2 = (2.54cm)^2 = 6.45cm^2
You know that you can always multiply by 1 without changing things. Dimensional analysis is just a way of multiplying by 1 that gets rid of the units you don't want, leaving behind the units you do want. It solves the problem of "what do I divide by?" or "do I multiply instead?"

You always multiply. By 1.

If 1in = 2.54cm, then divide by cm, and you get
1in/2.54cm = 1
similarly, 2.54cm/1in = 1

You want to convert from ft^2 to cm^2. I don't recall how many cm in a foot, but I do remember 2.54cm/in. So,

35.2ft^2 * (12in/ft)^2 * (2.54cm/in)^2 = (35.2*144*6.45)cm^2
You can check this at any of the many online unit conversion web sites.

The unwanted units cancel out, just as do common factors when reducing fractions to lowest terms. All we are left with is cm^2, as we wanted.