Why do we use $cm^2$?

Question

I can't seem to wrap my head around why we should use $cm^2$ for area.
According to my textbook we use it for converting units of area but I don't understand how $1cm$ is any different from $1cm^2$.

Can someone please explain why I should change the unit from $cm$ to $cm^2$ when working with area?

Answer 1

We use cm when we want to measure the length of a one dimensional object. Cm can be used when finding perimeter, or when finding the length of a line.

cm$^2$ is used when we want to find the specific area of something. Why cm$^2$? Let's say we want to find the area of a rectangle that is $8$cm long and $7$cm wide. Normally you would just do $8\times 7=56$ and attach the cm$^2$ to the end of it. But let's do this properly, using the actual units in our calculation. $$A=8\text{cm}\times 7\text{cm}=56\times \text{cm}\times \text{cm}=56\text{cm}^2$$ Why can't you use cm when finding area? Remember, cm is to find length. How would you find the length of the inside of this square? $$\square$$ You can try to draw lines in the square to find the length. But lines have no area$^*$ (which is why you can't use cm$^2$ for measuring length) so you could draw one billion lines and still be stuck in the first half. You will never be able to fill up that square using lines! So you cannot use cm for area$^{**}$.

$^*$ Technically the area of a line is $0$cm$^2$. Say you want to find the area of a line $8$cm long. This line has $0$cm width, so the area would be $8$cm $\times$ $0$cm $=$ $0$cm$^2$

$^{**}$ You can say that the length of the inside of a square is $\infty$ cm (it does not matter what unit you use actually).

Answer 2

They measure different things: $\mathrm{cm}$ is a measure of length, and $\mathrm{cm}^2$ is a measure of area (and $\mathrm{cm}^3$ is a measure of volume).

If a rectangle has side lengths $a\, \mathrm{cm}$ and $b\, \mathrm{cm}$ then the area of the rectangle is $(a \times b)\, \mathrm{cm}^2$.

Likewise, if a cuboid side lengths $a\, \mathrm{cm}$, $b\, \mathrm{cm}$ and $c\, \mathrm{cm}$ then the volume of the cuboid is $(a \times b \times c)\, \mathrm{cm}^3$.

Answer 3

This has 1 point

$$ \cdot $$

This has a length of 0 cm:

$$ \cdot $$

This has an area of 0 cm${}^2$:

$$ \cdot $$

This has $\infty$ points

$$ - $$

This has a length of approximately one cm:

$$ - $$

This has an area of zero cm${}^2$:

$$ - $$

This has $\infty$ points

$$ \blacksquare $$

This has a length of $\infty$ cm:

$$ \blacksquare $$

This has an area of approximately one cm${}^2$:

$$ \blacksquare $$