Cubic centimeters

Question

Simple question which applies to chemistry in a measurement context as i am trying to understand centimeters cubed. If we calculate a box's volume. The width, length and height of a box are $15.3, 27.3$ and $5.4 cm$, respectively.

So we take $(15.5)(27.3 )(5.4)$ and we get $2285.01$ as our volume. My book says it comes out to $2285.01cm^3$. Why do we write $cm^3$? To me this means that the box is $2285.01$ by $2285.01$ by $2285.01$ not 15.3 by 27.3 by 5.4? This makes sense for a $1cm^3$ as it is $1$ by $1$ by $1$ but for this specific example it doesn't make sense. Is the $cm^3$ simply to indicate that it has $3$ sides being times together ?

Answer 1

It's the conventional representation of a unit of volume.

A unit of length (one linear dimension): "$cm$".

A unit of area (two linear dimensions multiplied together: "$cm^2$" (which is actually equivalent to "$cm \times cm$").

A unit of volume (three linear dimensions multiplied together: "$cm^3$" (which is actually equivalent to "$cm \times cm \times cm $").

We stop there because we experience only $3$ spatial dimensions in our known Universe.

A unit cube ($1 cm \times 1 cm \times 1 cm$) has volume $1cm^3$.

When you say a cube has volume $2 cm^3$, you mean that it has the equivalent volume of the sum of two individual unit cubes. Not that it has the dimensions $2 cm \times 2 cm \times 2 cm$ (that would be a much larger cube with volume $8 cm^3$).

Answer 2

Your are correct that $1 \text { cm}^3$ corresponds to a cube $1 \text { cm}$ on a side. If you have a cube $3 \text { cm}$ on a side it takes $3^3=27$ of those little cubes to build it. The point is that when we say $27 \text { cm}^3$ we do not mean $(27 \text { cm})^3$ but $27 (\text{ cm})^3$. This carries over in real life. If you are told that a gallon of paint covers $200 \text { ft}^2$ you expect it to cover the floor of a $10 \times 20$ foot room, not a $200 \times 200$ foot room and if your room is $20 \times 40$ feet you need four gallons.

Answer 3

Consider the following picture:

Assume that this figure represents a $16\text{ cm} \times 4 \text{ cm}$ figure. What is its area? It is $16*4 = 64 \text{ cm}^2$...there are $64$, $1 \text{ cm}^2$ blocks that make up this area. It's not correct to say that there is $64 \text{ cm} \times 64 \text{ cm} = 4096 \text{ cm}^2$ that make up that space. The exact same $64 \text{cm}^2$ blocks can make up the following areas: