Area of a rectangle

The area of a figure says how much space it takes up on the plane. We measure it in little squares: the area is the number of unit squares that fit inside the figure. For a rectangle the answer is remarkably simple, because those squares line up in equal rows.

If the base is $a$ long, one row holds $a$ squares. There are as many rows as the height, that is $b$. Instead of counting them one by one, we multiply:

That is the whole formula for the area of a rectangle — and the reason multiplication looks like a rectangle in the first place.

Example: a 7 cm by 4 cm rectangle

The order of the multiplication makes no difference — $7 \cdot 4$ and $4 \cdot 7$ are the same. A rectangle turned a quarter turn has exactly the same area.

A square is a special rectangle

A square has all sides equal, so the same number goes into the formula twice:

That is where the name "squared" comes from: $a^2$ is read "$a$ squared" because it is literally the area of a square with side $a$. For $a = 5\,\text{cm}$ we get $A = 5^2 = 25\,\text{cm}^2$.

Area is not perimeter

These are two different quantities, and mixing them up is the most common mistake in this topic. The perimeter is the length of the line around the figure — the sum of all its sides:

So a $7\,\text{cm} \times 4\,\text{cm}$ rectangle has an area of $28\,\text{cm}^2$ but a perimeter of $2 \cdot (7 + 4) = 22\,\text{cm}$. Area comes from multiplying and is measured in square units; perimeter comes from adding and is measured in ordinary length units.

A useful check: "am I buying paint or skirting board?" Paint covers a surface — that is area. Skirting board runs around the edge — that is perimeter.

Finding a side from the area

The formula $A = a \cdot b$ works in the other direction too. Given the area and one side, the other side comes from a division:

Units of area

Area is written in square units because we multiply two lengths: $\text{cm} \cdot \text{cm} = \text{cm}^2$. One $\text{cm}^2$ is a little square with a side of $1\,\text{cm}$.

Converting them needs care: the length factor gets squared.

The side grows a hundred times, but the area grows a hundred times in each direction — ten thousand times over. That is why $1\,\text{m}^2$ is not $100\,\text{cm}^2$.

Before you compute an area, check one more thing: both sides must be in the same unit. A $2\,\text{m} \times 50\,\text{cm}$ rectangle is only computed after converting — $200\,\text{cm} \times 50\,\text{cm} = 10\,000\,\text{cm}^2$, which is $1\,\text{m}^2$.

Problems

Type the answer together with its unit, e.g. 28 cm² or 22 cm. That is part of the exercise: an area is written in square units, while a perimeter and a side length stay in ordinary units of length.

Common mistakes

• Confusing area with perimeter — area multiplies, perimeter adds; paint is area, skirting board is perimeter.
• Dropping the square unit — an area of $28\,\text{cm}^2$ is not $28\,\text{cm}$.
• Computing an area from sides in different units — convert first, multiply second.
• Turning $1\,\text{m}^2$ into $100\,\text{cm}^2$ — the length factor gets squared: $10\,000\,\text{cm}^2$.
• Multiplying all four sides — a rectangle has two pairs of equal sides, and the formula uses only two adjacent ones.
• Subtracting instead of dividing when finding a side — from $A$ and $b$ the other side comes out of $A / b$, not $A - b$.