Question

If one square mile is 640 acres, how many square meters are there in 1 acre?

Answer 1

Hint: First determine how many square miles are in 1 acre. Then using an empirical relation between kilometres (or metres) and miles, determine how many square meters are there in one square mile by squaring the said empirical relation. To this end, you will be able to establish a conjunction between acres and square metres to square miles, which will lead you to the appropriate result.

Formula Used:
$1\;mi = 1609\;m$

Complete Solution:
Let us begin by first understanding the units presented to us.
Miles and metres are units of distance. This means that the quantity that they measure represents the length/distance between any two given fixed points. When squared (or raised to the power of 2), these units become a measure of area.

This means that the quantity that $mi^2$ and $m^2$ measure represent the surface or two-dimensional space bound by sides of a given length. Thus, if the measurement of the length/distance between any two points was ‘$y$’ miles, then the area bounded by two sides of length $y$ miles is expressed as $y\times y= y^2\;mi^2$, where we multiplied the length two times. The same goes for metres as well.

Acres $(ac)$ are units of area in the British Imperial and US Customary Systems, a square mile ($mi^2$) is an English unit of area, whereas square metres ($m^2$) is the SI unit of area.

From the question, we thus deduce that we are given measures of area that need to first be converted from $ac$ to $mi^2$.
We are given that $640\;ac = 1\;mi^2 \Rightarrow 1\;ac = \dfrac{1}{640}\;mi^2$
We know that $1\;mi \approx 1.609\;km = 1.609 \times 10^3\;m = 1609\;m$

Squaring the above expression, we get:
$\Rightarrow (1\;mi)^2 = (1609\;m)^2$

Now, putting all our conversion relations together we get:
$1\;ac = \dfrac{1}{640}\;mi^2 = \dfrac{1}{640} \times (1609)^2\;m^2 \approx 4045\;m^2$

Therefore, there are approximately 4045 square metres in 1 acre.

Note:
It is advisable to remember the conversion relation between $mi \rightarrow m$ since it is a fundamentally empirical relation that cannot be mathematically derived. It is also important to distinguish between units of length and area and apply the appropriate relations by taking into consideration the units and their exponential powers that signify their dimensions. Always ensure that there is a consistency in the dimensionality of the units used so as to account for all conversion factors and their powers.