Area and Perimeter

What you will learn

• Tell the difference between area and perimeter.
• Find the area and perimeter of squares and rectangles.
• Split a compound shape into rectangles.

Before you start: you should know your multiplication facts and the shapes square and rectangle. See Shapes All Around.

The idea

Perimeter is the total distance around the edge of a shape. Walk round a field and the distance you walk is its perimeter. It is a length, measured in cm, m or km.

Area is the amount of flat space a shape covers. Cover the field with square tiles and the number of tiles is its area. It is measured in square units such as $\text{cm}^2$ or $\text{m}^2$.

Rectangles and squares

A rectangle with $l$ columns and $w$ rows of unit squares has $l \times w$ squares inside. Its perimeter is $2(l + w)$. A square is a rectangle with $l = w = s$, so $A = s^2$ and $P = 4s$.

Same perimeter, different area

A $1 \times 11$ rectangle and a $6 \times 6$ square both have perimeter $24$, but their areas are $11$ and $36$. For a fixed perimeter, the square encloses the most area of any rectangle.

Compound shapes

Split an L-shape into rectangles and add, or complete it to a rectangle and subtract the missing piece.

Worked examples

Example 1: Rectangle

A rectangular garden is $9$ m long and $4.5$ m wide. Find the length of fencing needed and the area of turf.

1. Perimeter: $2 \times (9 + 4.5) = 2 \times 13.5 = 27$ m.
2. Area: $9 \times 4.5 = 40.5 \text{ m}^2$.

Answer: $27$ m of fencing, $40.5 \text{ m}^2$ of turf.

Example 2: Square

A square patio has an area of $64 \text{ m}^2$. Find its perimeter.

1. The side is the number that multiplies by itself to give $64$, which is $8$ m.
2. $P = 4 \times 8 = 32$.

Answer: $32$ m

Example 3: Compound shape

An L-shaped room is made of a $6 \text{ m} \times 4 \text{ m}$ rectangle with a $3 \text{ m} \times 2 \text{ m}$ rectangle attached. Flooring costs £22 per square metre. What does it cost?

1. Split the L into two rectangles: $6 \times 4 = 24 \text{ m}^2$ and $3 \times 2 = 6 \text{ m}^2$.
2. Total area: $24 + 6 = 30 \text{ m}^2$.
3. Cost: $30 \times 22 = 660$.

Answer: £660

Tip: make sure all lengths are in the same unit before multiplying. A $2$ m by $50$ cm rectangle has area $1 \text{ m}^2$ or $10\,000 \text{ cm}^2$, but never $100$.

Why it matters

Paint, carpet, tiles, turf and fencing are all sold by area or length. Mixing up area and perimeter is an expensive mistake: buying fence panels by the square metre would leave you seriously short. Areas of other shapes come next in Area of Triangles, Parallelograms and Trapezia.