Area

In Grade 6 you met the area formulae for straight-sided shapes one at a time. In Stage 8 the goal is to use them fluently — picking the right one instantly and getting the answer without hesitation.

Here are the three you should now know by heart:

• Triangle: $\text{Area} = \tfrac{1}{2} \times \text{base} \times \text{height}$
• Parallelogram: $\text{Area} = \text{base} \times \text{height}$
• Trapezium: $\text{Area} = \tfrac{1}{2} \times (a + b) \times h$

The thread running through all three is the perpendicular height — the straight-up distance, at right angles to the base. The slanting side is never the height.

A reliable habit: write the formula down first, then substitute the numbers, then calculate. That tiny extra step stops you forgetting to halve a triangle, or doing the trapezium bracket too late.

A circle is your first curved shape for area, so it is worth being clear about its parts.

• The centre is the point exactly in the middle.
• The radius ($r$) is the distance from the centre to the edge.
• The diameter ($d$) is the distance straight across, through the centre — so $d = 2r$.

Why does this matter so much? Because the area formula uses the radius, not the diameter. If a problem gives you the diameter, your very first move is to halve it.

You should also know the special number π (pi). It is the number you get when you divide any circle's circumference by its diameter — it is always the same, roughly $3.14$ (or you can use the π button on a calculator).

Get comfortable spotting which measurement you have been given. A quick label on the diagram — "this is $r$" — saves a lot of mistakes later.

Here is the headline formula of Stage 8 area work: $\text{Area} = \pi \times r^2.$

The order matters. The little ² applies only to the $r$, so you square the radius first, and only then multiply by π.

For a circle of radius $5\text{ cm}$: $\text{Area} = \pi \times 5^2 = \pi \times 25 \approx 78.5\text{ cm}^2.$

Watch what happens if you do it in the wrong order: $\pi \times 5 = 15.7$, and then squaring gives a completely different (and wrong) answer. Square first, multiply second.

A few tips:

• If you are given the diameter, halve it to get $r$ before anything else.
• Use $\pi \approx 3.14$, or the π button, and round your final answer sensibly.
• The answer is an area, so it is in square units — $\text{cm}^2$, $\text{m}^2$.

For a semicircle (half a circle), find the full circle area and then halve it.

A compound shape is built from simpler shapes joined together. In Stage 8 these get more interesting — they often include a triangle, a trapezium, or even a semicircle, not just rectangles.

The method stays the same: split, find each piece, then add (or sometimes subtract).

A clear plan for any compound shape:

• Draw your splitting line and label each piece.
• Work out any missing lengths by subtracting known lengths.
• Find each area with the right formula, keeping units the same.
• Add the pieces — or, for a shape with a hole, find the whole and subtract the hole.

Sometimes the quickest route is "whole minus hole": imagine the shape complete, find that area, then subtract the missing part. For example, a square with a circle cut out has area $(\text{square}) - (\pi r^2)$.

Take it slowly — neat labelling is what stops these going wrong.

Stage 8 problems often turn things around: instead of giving you the lengths, they give you the area and ask for a missing length.

The trick is to put what you know into the formula, then rearrange to find the unknown.

For a rectangle with area $48\text{ cm}^2$ and width $6\text{ cm}$: $\text{length} = \text{area} \div \text{width} = 48 \div 6 = 8\text{ cm}.$

For a triangle with area $30\text{ cm}^2$ and base $10\text{ cm}$, undo the formula step by step: $\tfrac{1}{2} \times 10 \times h = 30 \;\Rightarrow\; 5h = 30 \;\Rightarrow\; h = 6\text{ cm}.$

You can even work backwards with a circle: if a circle has area $\approx 50.24\text{ cm}^2$, then $r^2 = 50.24 \div \pi \approx 16$, so $r = \sqrt{16} = 4\text{ cm}$.

The skill here is staying calm and undoing the formula one operation at a time — exactly the inverse-operations idea you use when solving equations.

The area skills you have sharpened this year carry straight into harder topics:

• Surface area — the surface area of any solid is the total of the areas of its faces.
• Volume of prisms and cylinders — volume is a cross-section area multiplied by a length, and a cylinder's cross-section is a circle.
• Circumference — the area of a circle pairs naturally with finding the distance around a circle.
• In everyday life, area decides how much paint, turf, carpet or material a job needs.

If a later measure topic feels hard, an area skill is often underneath it. Keep this guide close — confident, careful splitting gets easier every single time.