# Proof 1: Rectangle, square, box and cube area and volume

A room is $4\,\mathrm m$ long and $2\,\mathrm m$ wide’; a rectangle-shape. For each of the 4 metres there are 2 square metres across. That is 2 square metres 4 times. That is $8\,\mathrm m^2$. Length $l$ and width $w$ are multiplied together:

$A_{\text{rectangle}}=lw\quad_\blacksquare\quad$ 1

The room is also $3\,\mathrm m$ high’; it is now a box-shape. There are 3 cubic metres above every square metre floor. That is 8 times that we have 3 cubic metres; in total $24\,\mathrm m^3$. Area of the bottom or base is multiplied with the height $h$:

$$V_{\text{box}}=A_{base}h=lwh \quad_\blacksquare$$

A room could easily be just as long as it is wide – that would be a square – and also just as high – making it a cube. The floor still has rectangle-shape and the room still box-shape but with the special case that $l$, $w$ and $h$ are equal. It doesn’t matter which edge we call ‘length’, ‘width’ and ‘height’. Let’s just call them all the same, such as $s$ (for ‘side’):

$$A_{\text{square}}=ss=s^2 \quad_\blacksquare$$

$$V_{\text{cube}}=sss=s^3 \quad_\blacksquare$$

References:

1. I Want to be a Mathematician’ (book), Paul R. Halmos, Springer-Verlag New York, 1st ed., 1985, www.springer.com/la/book/9780387960784, ISBN 978-0-387-96470-6, DOI 10.1007/978-1-4612-1084-9
2. Measure Theory’ (book), Paul R. Halmos, Springer-Verlag New York (originally by Litton Educational Publishing), 1st ed., vol. 18, 1950, www.springer.com/la/book/9780387900889, ISBN 978-1-4684-9442-6, DOI 10.1007/978-1-4684-9440-2
3. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com