How to convert cm² to km²?

I have a problem that says "a small hole is $20.7~\mathrm{cm^2}$ wide. What is the area in $\mathrm{km^2}$."

I started doing it like this:

$20.7~\mathrm{cm^2} \cdot 0.01~\mathrm{m^2} / 1~\mathrm{cm^2} = 0.207$

$0.207 \cdot 1~\mathrm{m^2} / 1000~\mathrm{km^2} = 0.000207$ or $2.07\cdot10^{-4}$

My answer says it should be $10^{-9}$. What am I missing here?

• This question appears to be off-topic because it belongs to Mathematics S.E. – ashu Sep 4 '14 at 3:28
• @ashu Unit transformations are a large component of Chemistry and Chemistry related analyses. – LordStryker Sep 4 '14 at 13:55

1 square centimeter = 0.0001 square meters

$$(20.7\ \mathrm{cm}^2) \left ( \dfrac{0.0001\ \mathrm{m}^2}{1\ \mathrm{cm}^2} \right ) = 2.07 \times 10^{-3}\ \mathrm{m}^2$$

1 square meter = 1 x 10^(-6) square kilometers

$$(2.07\times10^{-3}\ \mathrm{m}^2) \left ( \dfrac{1\times10^{-6}\ \mathrm{km}^2}{1\ \mathrm{m}^2}\right) = 2.07 \times 10^{-9}\ \mathrm{km}^2$$

• Just to be I'm getting this right and it's not just a coincidence, is it the normal conversion (like cm to m is -2, and m to km is -3) times the unit's exponent? So since it's squared, it's say cm^2 so -2*2, (4), and -3*2, (-6), for the other? – Caesium-133 Sep 3 '14 at 22:04
• Exactly, you've got it! – ron Sep 3 '14 at 22:07
• Ok, so in that one I just had 20.7. if it was 20.7 x 10^(-5), or any exponent in sci. notation form adding in, what am I doing with that part? Adding the combo of the conversion and unit exponents to the one in the sci. notation? Or? – Caesium-133 Sep 3 '14 at 22:28
• When you multiply powers of 10, you add the exponents, so 20.7 x 10^(-4) x 10^(12) = 20.7 x 10^8. Is that what you're asking about? – ron Sep 3 '14 at 22:39
• Like 1.36 x10^(-6)cm^3 to m^3 – Caesium-133 Sep 3 '14 at 22:47