Why do numbers not change when they become a square or cubed number in measurement

Question

For example,I'm trying to find the volume of a square with a side length of $3cm$. When we find the volume of $27$, it is written $27cm^3$. What I'm confused about is, why does the number not change when it becomes $cm^3$ or $cm^2$. When you convert $cm^2$ or $cm^3$ between each other, you need to multiply it by a particular number. I don't get why it's not the case for this.

Answer 1

You are muddling up the different ways that measurements are converted/calculated.

If you convert from cm to m then you divide by 100. This is because the 'c' means centi or $\frac{1}{100}$th of the measurement. This is true for any prefixes we use in front of a measurement.

What you are talking about is converting from cm$^2$ to cm$^3$ which isn't possible as they aren't the same thing. The first measures an area whereas the second measures a volume. When you find the volume of a square with side length of 3cm you are performing a calculation not a conversion. The calculation you are doing is 3cm$\times$3cm$\times$3cm which you can work out just by multiplying. Multiplying the numbers gives the 27 while multiplying the units give cm$^3$.

Answer 2

You are mixing things.

Let you have square with side 3 cm.

Then volume = (side)$^3$

Volume = (3 cm)$^3$

= 27 cm$^3$

Now suppose you have volume of a square then you can't find area of square directly. You have to first find side then find area.

Let Volume = 64 cm$^3$

Then side = $(64 cm^3)^{\frac{1}{3}}$

side = 4 cm

Now area = (side)$^2$

= $(4 cm)^2$

= 16 cm$^2$