Unit Conversions  Example #7 – Square Units

Here is another situation for unit conversion
that is sometimes confusing to people. The specific question here is converting 500
inches squared (or square inches) into centimetres squared (or square centimetres depending on
what you call it). The issue here is converting units with exponents
in them. Both inches squared and centimetres squared
are units for measuring surface area, and generally we use units of length with an exponent
of 2 to measure areas just like we use units with an exponent of 3 to measure volumes of
three-dimensional spaces. I’m going to show you a mathematical method
for solving this problem that you can apply to any other problem with exponents on units. So, we are starting with 500 inches squared
and we want to know what the area would be in centimetres squared. The first thing we need is a relationship
between inches and centimetres. Now, you might ask at this point, “Why don’t
we find a relation between inches squared and centimetres squared, if those are the
units we’re actually converting between. And the answer is, you could do this question
that we, if you had a relationship like that available, but I’m going to teach you this
more general method because I think it is more useful, it’s not difficult and you can
use this method with any squared unit or cubed unit or even units with higher exponents,
like say if you had metres to the power of four, if you ever ran into that kind of question.” But anyway, I know the relationship is 1 inch
is equal to 2.54 centimetres. From this relation comes the standard two
conversion factors, 1 inch over 2.54 centimetres and 2.54 centimetres over 1 inch. And I want to use the one with inches on the
bottom so it will cancel. That’s because inches is on the top in the
original amount if you write it as a fraction. So far, this is exactly the same as any unit
conversion problem, but here is where things are going to change. Because my original amount was in inches squared,
I have to square the entire conversion factor before I multiply by it. So I’ll put this whole fraction to the exponent
two, and this is what it looks like. Now, it helps here to remember a little about
how fractions work with exponents. Because the exponent is outside the bracket,
this is, arount the entire fraction, this exponent applies to the entire fraction top
and bottom. So I need to square both the top and the bottom
of this fraction, and the way it works out is the 2.54 squared is 6.45 and centimetres
squared is just centimetres squared, because that’s what happens when you square a unit
or raise a unit to any exponent, really, it’s just… the unit gets put to that exponent. So, what I just did was take this whole top
– 2.54 cm – and it because 2.54 squared and centimetres squared – the 2.54 squared is
6.45 and the centimetres squared is just centimetres squared. And now, doing the same thing with the bottom
of the fraction, I get 1 squared is 1 and inches squared is just inches squared. So, I did bottom the exact same way as the
top, and this is the result. So now, you can see with these two fractions
that inches squared will cancel, and all we have left to do is mulitply the fractions
together. 500 times 6.45 over 1 is equal to 3226, and
the only unit that hasn’t been canceled is centimetres squared, so that’s the answer. So, I’ll just write my final statement stating
that 500 inches is the same as 3226 centimetres squared. And that is the correct answer, but it’s always
a good idea to check and make sure it makes sense. Does it make sense that over 3000 centimetres
squared covers the same area 500 inches squared? And if you don’t have an idea of how big those
areas are, just check over each step, make sure the units cancelled properly (since if
they didn’t cancel, that could be a source of error), and make sure all the multiplying
was done right.

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