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 \( \color{red}{\textbf{false}}\).

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 \( (1\text{ m})^2 = (100\text{ cm})^2 = 100^2 \text{ cm}^2 \).