# 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:

- ‘
**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 - ‘
**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 - ‘
**Online Etymology Dictionary**’ (dictionary),*Douglas Harper*, www.etymonline.com

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