Area of a Rectangle: Length Times Width

Counting squares

Area is how much space is inside a shape.

It is counted in unit squares — squares one unit wide and one unit tall.

Cover this rectangle in $1 \times 1$ squares. Each row holds $8$ of them, and there are $5$ rows:

$$5 \text{ rows} \times 8 \text{ per row} = 40 \text{ squares}$$ $$A = l \times w$$

Why multiplying works

The formula is not a rule to take on trust. It is what counting the squares turns into.

Multiplying is how you count equal groups. The rows of a rectangle are equal groups. So “five rows of eight” is $5 \times 8$, for the same reason “five bags of eight apples” is.

It also holds when the sides are not whole numbers. A rectangle $2.5$ by $4$ has half squares in its last row. Two halves make a whole, so the count still works: $2.5 \times 4 = 10$.

Why the units are squared

Each tile is one centimetre wide and one centimetre tall. That is one square centimetre. Count $40$ of them and you have $40$ square centimetres, written $40$ cm².

The small $2$ is not decoration. It says that two lengths were multiplied. It is also how you tell an area from a perimeter at a glance.

Quantity	Multiplied	Units
length	—	cm
area	two lengths	cm²
volume	three lengths	cm³

Squares

A square is a rectangle whose sides are equal, so

$$A = s \times s = s^2$$

That is where the word “squared” comes from. $s^2$ is the area of a square with side $s$.

Working backwards

Area divides as well as it multiplies. Know the area and one side, and you can find the other:

$$A = l \times w \quad\Rightarrow\quad w = \frac{A}{l}$$

A rectangle with area $40$ cm² and length $8$ cm has width $40 \div 8 = 5$ cm.

L-shapes: cut and add

Areas add up. So cut an awkward shape into rectangles and add the pieces.

A room shaped like an L splits into a $6 \times 4$ rectangle and a $3 \times 2$ rectangle:

$$24 + 6 = 30 \text{ m}^2$$

No new formula is needed. Cut it wherever suits you. The total comes out the same.

Worked examples

Common mistakes

Practice problems