Area

Area measures the amount of flat surface a shape covers. Think of how much paint you would need to colour it in, or how many floor tiles would cover a room.

We measure area by counting squares. A square that is $1\text{ cm}$ along each side has an area of one square centimetre, written $1\text{ cm}^2$.

The rectangle above is 4 squares wide and 3 squares tall, so it covers $4 \times 3 = 12$ squares. Its area is $12\text{ cm}^2$.

A few things to remember from the start:

• Area is always given in square units — $\text{cm}^2$, $\text{m}^2$, $\text{mm}^2$.
• This is different from perimeter, which is the distance all the way around.
• Counting squares works for any shape, but formulae let you find area much faster.

For a rectangle, you do not need to count every square. The squares form neat rows, so you just multiply: $\text{Area} = \text{length} \times \text{width}.$

So a rectangle that is $8\text{ cm}$ long and $5\text{ cm}$ wide has area $8 \times 5 = 40\text{ cm}^2$.

A square is just a rectangle whose length and width are equal, so: $\text{Area} = \text{side} \times \text{side}.$

A square with side $6\text{ cm}$ has area $6 \times 6 = 36\text{ cm}^2$.

Two quick tips:

• Check both measurements are in the same units before you multiply. Change $1\text{ m}$ to $100\text{ cm}$ if needed.
• The order of multiplication does not matter — $8 \times 5$ and $5 \times 8$ both give 40.

This simple formula is the building block for every other area formula, so it is well worth knowing inside out.

Every triangle is exactly half of a rectangle that surrounds it. That gives the formula: $\text{Area} = \tfrac{1}{2} \times \text{base} \times \text{height}.$

So a triangle with base $10\text{ cm}$ and height $6\text{ cm}$ has area: $\tfrac{1}{2} \times 10 \times 6 = 30\text{ cm}^2.$

The most important word here is perpendicular height — the height has to be measured at right angles to the base, straight up from it. It is not the length of a slanting side.

A handy order of working:

• First multiply the base by the height.
• Then halve the result.

This formula works for any triangle — right-angled, isosceles or completely lopsided — as long as you use a perpendicular height that matches the base you chose.

A parallelogram is a four-sided shape with two pairs of parallel sides — like a rectangle that has been pushed over to one side.

If you cut a right-angled triangle off one end and slide it to the other, the parallelogram turns into a rectangle. That is why the formula looks so familiar: $\text{Area} = \text{base} \times \text{perpendicular height}.$

So a parallelogram with base $9\text{ cm}$ and perpendicular height $5\text{ cm}$ has area: $9 \times 5 = 45\text{ cm}^2.$

Notice there is no halving here — that step belongs to triangles. A parallelogram with the same base and height as a triangle has exactly double its area.

Once again, the height must be perpendicular to the base. The slanted side is usually longer, and using it would give an answer that is too big.

A trapezium is a four-sided shape with one pair of parallel sides. Its area formula looks longer, but it follows a clear idea.

If the two parallel sides have lengths $a$ and $b$, and the perpendicular distance between them is $h$, then: $\text{Area} = \tfrac{1}{2} \times (a + b) \times h.$

The clever part is $\tfrac{1}{2}(a + b)$ — that is just the average of the two parallel sides. So the formula says: find the average width, then treat the trapezium like a rectangle of that width.

For the trapezium above, with $a = 6$, $b = 12$ and $h = 5$: $\tfrac{1}{2} \times (6 + 12) \times 5 = \tfrac{1}{2} \times 18 \times 5 = 45\text{ cm}^2.$

Always do the bracket first (add the parallel sides), exactly as BIDMAS tells you. Only the two parallel sides go inside the bracket — the slanting sides are not used.

A compound shape is made by joining simple shapes together — like an L-shape or a house outline. There is no single formula for it, but there is a reliable method.

Split the shape into rectangles and triangles, find each area, then add them up.

For example, suppose the L-shape splits into:

• Rectangle A: $7\text{ cm} \times 14\text{ cm} = 98\text{ cm}^2$.
• Rectangle B: $9\text{ cm} \times 6\text{ cm} = 54\text{ cm}^2$.

Total area $= 98 + 54 = 152\text{ cm}^2$.

A few tips for these:

• Draw your splitting lines lightly and label each piece.
• You often have to work out a missing length by subtracting one length from another.
• Sometimes it is easier to find a big rectangle and subtract a missing corner.
• Check every piece is in the same units before adding.

Getting confident with area now sets you up for plenty ahead:

• Surface area — the surface area of a 3D solid is found by adding up the areas of all its faces.
• Volume — the volume of a prism is its cross-section area multiplied by its length.
• Circles — later you will meet the area of a circle, $\pi r^2$.
• In everyday life, area is used for flooring, painting, gardening, buying carpet and planning rooms.

If a measurement topic later feels hard, it is often an area skill underneath that needs a refresh. Keep this guide nearby — splitting shapes carefully gets easier every time you do it.