Converting squared or cubed units

Question

At the moment I am going from square meters to square yards. One meter is $1.0936$ yards.

So I figure $1\text{m}^2 = 1.0936 \text{yards}^2$

I know it isn't but I want to learn a good systematic way to understand why I square each part of the yards side i.e $1.0936^2\text{yards}^2$ even though I did not square each part of the other side.

Answer 1

Since we've got

$1 \text{m} = 1.0936 \text{yd}$

we can rearrange to conclude that

$$1 = \frac{1.0936 \text{yd}}{1 \text{m}}$$

Thus, we have

$$1 \text{m}^2 = \left(1 \text{m}^2\right) \cdot 1 \cdot 1 = \left(1 \text{m}^2\right) \frac{1.0936 \text{yd}}{1 \text{m}} \cdot \frac{1.0936 \text{yd}}{1 \text{m}} = (1.0936)^2 \text{yd}^2$$

We had to multiply by the conversion factor twice, since we have the unit m twice.

Answer 2

The number $1.0936$ is a little hard to grasp. So let us go from square yards to square feet.

Recall that $1$ yd is equal to $3$ feet.

Now what is $1$ yd$^2$? By definition it is the area of a $1$ yd by $1$ yd square.

Divide that square into squares $1$ foot by $1$ foot. You can do that by drawing $2$ horizontal lines and $2$ vertical lines. Do the actual drawing. You will end up with $9=3^2$ little squares that are $1$ foot by $1$ foot. So your area is $3^2$ square feet.

Thus to convert from square yards to square feet, you multiply by $3^2$.

Analogously, to convert from square metres to square yards, you multiply by $(1.0936)^2$.

Similarly, imagine a cube with sides $1$ yd. If you divide it into smaller cubes with each side $1$ ft long, you will end up with $27=3^3$ cubes. Thus $1$ cubic yard is $27$ cubic feet.

Analogously, $1$ cubic metre is $(1.0936)^3$ cubic yards.