Units of area

Area is inherently two-dimensional. Length and perimeter are one-dimensional — measured directly in metres or centimetres. Area is a product of two lengths, so its unit is a square unit: $\text{cm}^2$ for smaller figures like a notebook page or a table top, and $\text{m}^2$ for larger ones like a room, a plot, or a field. This is why an area answer written as "$30\ \text{m}$" (missing the square) is always wrong, even if the number is right.

Why the conversion factor gets squared. Suppose the linear relationship is $1\ \text{m} = 100\ \text{cm}$. A square of side $1\ \text{m}$ has the same area as a square of side $100\ \text{cm}$ — but you cannot get from one number to the other by multiplying by $100$ once, because both the length and the breadth of that square grow by a factor of $100$ when re-expressed in centimetres:

where: the left side is one square metre, and the squaring on the right comes directly from squaring the linear factor $100$, since area is length $\times$ length.

The general scaling rule. If every linear dimension of a shape is scaled by a factor $k$, its area scales by $k^2$. This is the same "squaring effect" that makes doubling a rectangle's length and breadth quadruple its area — and it is exactly why unit conversions for area always involve squaring the linear factor, whatever the pair of units. Since $1\ \text{km} = 1000\ \text{m}$, for instance, $1\ \text{km}^2 = 1000^2\ \text{m}^2 = 10,00,000\ \text{m}^2$, even though this chapter's problems mostly stay within $\text{cm}^2$ and $\text{m}^2$.

One rule of thumb saves memorising a table. You never need to memorise separate area-conversion factors for every pair of units — remember the linear conversion, then square it. To go from $\text{m}^2$ to $\text{cm}^2$, multiply by $10{,}000$; to go from $\text{cm}^2$ to $\text{m}^2$, divide by $10{,}000$.

A rectangular plot is $12\ \text{m}$ long and $250\ \text{cm}$ wide. Find its area in $\text{m}^2$, and verify the answer by working entirely in $\text{cm}^2$ instead.

Step 1. Convert both dimensions to the same unit. Working in metres: $250\ \text{cm} = 250 \div 100 = 2.5\ \text{m}$.

Step 2. Apply the rectangle-area formula in metres: $A = l \times b = 12 \times 2.5 = 30\ \text{m}^2$.

Step 3. Verify using centimetres instead. Convert the other dimension: $12\ \text{m} = 1200\ \text{cm}$. Then $A = 1200 \times 250 = 3{,}00{,}000\ \text{cm}^2$.

Step 4. Convert this back to $\text{m}^2$ using the squared factor: $3{,}00{,}000 \div 10{,}000 = 30\ \text{m}^2$. Both routes agree.

The plot's area is $30\ \text{m}^2$.

Competitive stretch

A square tile has side $150\ \text{cm}$. How many such tiles are needed to cover a hall floor of area $54\ \text{m}^2$, with no wastage? One tile's area is $150 \times 150 = 22{,}500\ \text{cm}^2$. Converting the hall's area to the same unit, $54\ \text{m}^2 = 54 \times 10{,}000 = 5{,}40{,}000\ \text{cm}^2$. Dividing, $5{,}40{,}000 \div 22{,}500 = 24$ tiles.

• Converting an area unit by multiplying or dividing by the linear factor only once (using $100$ instead of $10{,}000$ between $\text{m}^2$ and $\text{cm}^2$) — the factor must be squared.
• Adding or comparing areas given in different units without converting both to a common unit first.
• Converting only one of a rectangle's two dimensions to a common unit before multiplying, leaving the other in a different unit.
• Writing an area answer with a plain linear unit, such as "$30\ \text{m}$," instead of the correct square unit "$30\ \text{m}^2$."

Ask the tutor: If unit conversions in area problems still trip you up, ask the AI tutor for a few quick practice conversions between $\text{cm}^2$ and $\text{m}^2$ to build confidence.