Does 1 square meter equal 100 square centimeters?

This is part of a series on common misconceptions.

Is $1\text{ m}^{2}=100\text{ cm}^{2}\, ?$

Why some people say it's true: $1\text{ m}=100\text{ cm},$ and therefore $1\text{ m}^{2}=100\text{ cm}^{2}.$

Why some people say it's false: Since $1\text{ m}=100\text{ cm}, 1\text{ m}^{2}=(100\text{ cm})^{2}\neq100\text{ cm}^2.$

The statement $“1\text{ m}^{2}=100\text{ cm}^{2}"$ is $\color{#D61F06}{\textbf{false}}$.

Proof:
$1\text{ m}^2$ represents the area of a $100\text{ cm}$ by $100\text{ cm}$ square, or a square composed of $100^{2}$ $1\text{ cm}$ by $1\text{ cm}$ squares. Hence, the area is $100^{2}\text{ cm}^2.$

We can also take a conversion approach. $1 \text{ m}^2 = 1 \text{ m}^2 \cdot \frac{100 \text{ cm}}{1 \text{ m}} = 100 \text{ m cm}.$ Here, one of the instances of $\text{m}$ in the $\text{m}^2$ will cancel, but another instance of $\text{m}$ will remain. So we must apply the conversion factor again:

$$100 \text{ m cm} \cdot \frac{100 \text{ cm}}{1 \text{ m}} =10,000 \text{ cm}^2 .$$

This can also be seen geometrically. Consider a 1-meter long stick. Because $1\text{ m}=100\text{ cm},$ we can mark off $100\text{ cm}$ units along the stick. $1\text{ m}^{2}$ can be visualized as a $1\text{ m} \times 1\text{ m}$ area. As you can see in the image below, this area is filled with $100 \times 100 \text{ cm}^2$ square areas. Therefore, $1\text{ m}^{2} =10,000 \text{ cm}^2 .$

Rebuttal: But when you make $\text{m}$ to $\text{m}^{2}$, all you do is $\text{m} \times \text{m} = \text{m}^{2}$. The same for $\text{cm}.$ So all you do is $1~ (\text{m} \times \text{m})=100~ (\text{cm} \times \text{cm})$.

Reply: $1\text{ m}^{2}=1\text{ m} \times 1\text{ m},$ while $1\text{ cm}^{2}=1\text{ cm} \times 1\text{ cm}$. We have to replace the $1\text{ m}$ by $100\text{ cm}$ in both instances. A simple way of remembering this approach is to compare it to the rule of exponent $(ab)^2 = a^2 b^2$, and hence $(1\text{ m})^2 = (100\text{ cm})^2 = 100^2 \text{ cm}^2$.