# Unit conversion multiplication

This is going to sound like a really dumb question. At first I thought it was just my brain failing to function, but I can't work my mind around it. I'm using a problem to demonstrate the conceptual issue, but what I care about is why things work however they may.

Let's say we have a rectangle $5~\text{cm} \times 10~\text{cm}$. What is the area in meters?

If we say it is an area of $50~\text{cm}$, our answer should be $.05~\text{m}^2$.

On the other hand, if we were to convert to meters first, $.05~\text{m} \cdot .1~\text{m} =.005~\text{m}^2$.

• Do you mean a rectangle with side lengths $5~\text{cm}$ and $10~\text{cm}$. Squares have congruent sides. – N. F. Taussig Mar 6 '16 at 3:08

There are $100~\text{cm}$ in a meter. A square meter is the area of a square with side length $1~\text{m} = 100~\text{cm}$. Hence, $$1~\text{m}^2 = (1~\text{m})(1~\text{m}) = (100~\text{cm})(100~\text{cm}) = 10000~\text{cm}^2$$ Thus, to convert an area given in $\text{cm}^2$ to an area given in $\text{m}^2$, we must multiply by the conversion factor $$\frac{1~\text{m}^2}{10000~\text{cm}^2}$$ Hence, in your example, the area in square meters of a rectangle with area $50~\text{cm}^2$ is $$50~\text{cm}^2 \cdot \frac{1~\text{m}^2}{10000~\text{cm}^2} = 0.005~\text{m}^2$$
i think you're underestimating how many $cm^2$ there are in a $m^2$.
50 $cm^2 * \frac{1 m^2}{10,000 cm^2} = .005 m^2$