Area & Perimeter

• Perimeter is the total distance around the outside of a 2D shape — just add up every side length.
• The perimeter of a circle has a special name: the circumference.
• Compound shapes (irregular ones) can be split into rectangles, triangles, and parts of circles.
• Use shape properties (equal sides, opposite sides equal, etc.) and sometimes Pythagoras' theorem to find missing lengths before you can find the perimeter.
• Area is the amount of 2D space inside a shape's boundary, measured in squared units (cm², m², etc.).
• You need to know the area formulas for a rectangle, triangle, trapezium, and parallelogram by heart (only the triangle one is given to you in the exam).
• To find the area of a compound shape: split it into standard shapes and add the areas — or sometimes it's easier to complete a bigger shape and subtract the extra bit.
• Problem-solving questions dress area up in real-life contexts (carpets, painting, tiling) — often involving cost and compound units like £/m².

Perimeter is simply the total distance you'd walk if you traced all the way around the outside edge of a 2D shape and ended up back where you started. Because it's a length, it's measured in "one dimension" — units like mm, cm, or m (never squared).

Think of a rectangular garden with a fence around it. The perimeter is the total length of fencing you'd need to buy. It doesn't matter what shape the garden is — you're just measuring the boundary line.

For any regular shape (all sides equal — like a square or equilateral triangle), you don't need to add up every side one by one. Just multiply the number of sides by the length of one side.

A compound shape is made of two or more simpler 2D shapes joined together (like an L-shape, which is really two rectangles pushed together). The strategy is always the same:

1. Look for sides that must be equal — dashes on a diagram, or the question telling you directly, often signal this (e.g. two sides of an isosceles triangle).
2. Use the shape's structure to work out missing lengths. A classic trick: on an L-shape, the sum of the two shorter horizontal (or vertical) sides equals the one long side directly opposite them.
3. Sometimes you'll need Pythagoras' theorem to find a missing side on a right-angled triangle hiding inside the compound shape.
4. Once every side length is known, just add them all up.

Area is the amount of flat space enclosed inside the boundary of a 2D shape — think of it as "how many 1cm × 1cm squares would it take to completely cover this shape with no gaps or overlaps?" That's why area is always measured in squared units: mm², cm², m², and so on. Unlike perimeter (a 1D length), area uses lengths in two dimensions multiplied together.

Before using formulas, it helps to understand area at its most basic level: counting squares.

• Count every whole square fully inside the shape.
• For partial squares, try to pair up halves (or other fractions) to make whole squares.
• If the grid has a scale (e.g. each square = 4 cm²), multiply your square count by that scale to get the total area.

These four formulas are the backbone of the entire chapter. You need to know all of them from memory — only the triangle formula is printed in the exam.

Forgot a formula in the exam? No panic — you can always split the trapezium or parallelogram into a rectangle and triangle(s) and work it out from the shapes you definitely remember.

A compound shape is any shape that isn't one of your standard shapes (rectangle, triangle, trapezium, parallelogram). It's built by combining two or more standard shapes together. Since you don't have a direct formula for an "L-shape" or a "pentagon-ish house shape," you break it down into pieces you have formulas for.

Strategy 1 — Split and Add: Cut the compound shape into standard shapes (usually rectangles and triangles), find each area, then add them all together.

Strategy 2 — Complete and Subtract: Sometimes it's actually easier to imagine the shape as a "bigger, simpler shape" with a chunk missing. Complete the bigger shape, find its area, then subtract the area of the missing chunk.

A pentagon looks like a rectangle with a triangular "roof" slanting up on one side. The rectangle part is 12 cm wide and 4 cm tall. Above it, the shape rises further: the left corner of the roof starts 5 cm up (matching a step), and the tallest point on the right reaches 9 cm total height, over a horizontal run — the full top-right point is 12 cm across from the left edge.

Step 1: Split into a rectangle (12 × 4) and a triangle sitting on top.

Step 2: Find the triangle's base and height using subtraction:

• Triangle height = 9 − 4 = 5 cm
• Triangle base = 12 − 5 = 7 cm

Step 3: Add the two areas:

These questions wrap area calculations inside a real-life story (painting a room, laying carpet, designing a field) and usually mix in a second topic — most commonly money/cost calculations, sometimes percentages. They tend to feel harder than they are because there's a lot of text to wade through before you reach an actual number.

• Don't panic if you can't immediately see the final answer — instead, ask "what can I calculate from what I've been given?" One calculation often unlocks the next.
• Watch for key words: "minimum," "maximum," "cheapest," "at least" — these tell you what kind of answer you're aiming for.
• Watch for compound units like £/m² — this is a huge clue that you'll need to multiply a cost-per-square-metre rate by an area.
• Annotate the diagram in pencil as you work out missing lengths — future-you (a few steps later) will thank present-you.
• Even if you don't reach a final answer, write down every calculation you attempt — exam mark schemes usually award method marks for correct working, even without a final correct number.

John's L-shaped lounge: one rectangle is 3.2 m × 3.6 m, and a smaller rectangle attached to it is 2.4 m × 1.8 m (found via 6 − 2.4 = 3.6 for the missing length). Two companies quote:

• Company A: Fixed £5.50 per m²
• Company B: £6 per m² for the first 10 m², then £4 per m² for anything over that

Step 1 — Find the area (split into two rectangles and add):

Step 2 — Cost with Company A:

Step 3 — Cost with Company B (split into "first 10 m²" and "remaining"):

Conclusion: John should choose Company B — it's £3.76 cheaper.

• I can explain the difference between perimeter and area
• I know the perimeter of a circle is called the circumference
• I can find the perimeter of a regular shape using P = n × s
• I can find missing side lengths on a compound shape (e.g. using the "two shorter sides = one longer side" rule)
• I can use Pythagoras' theorem to find a missing length when needed
• I can count squares on a grid to estimate area, including partial squares
• I know and can apply the area formula for a rectangle (A = l × w)
• I know and can apply the area formula for a triangle (A = ½bh)
• I know and can apply the area formula for a trapezium (A = ½(a+b)h)
• I know and can apply the area formula for a parallelogram (A = bh)
• I understand that "height" always means perpendicular height, not a slanted side
• I can split a compound shape into standard shapes and add their areas
• I can complete a shape into a bigger one and subtract the extra area when it's easier
• I can solve real-life word problems involving area and cost (e.g. carpets, paint)
• I can spot and use compound units like £/m² in problem-solving questions

• Mixing up perimeter and area units. Perimeter answers should never have a "²" — if you see cm² in a perimeter answer, something's gone wrong.
• Using the slanted side instead of the perpendicular height for triangles and parallelograms. Always look for the right-angle marker in the diagram to confirm which length is the true height.
• Forgetting to halve in the triangle and trapezium formulas — it's an easy slip under exam pressure. Write the ½ down first, before you plug in numbers.
• Not finding missing lengths before calculating. Compound shape questions almost always require a subtraction step (like 17 − 3 = 14) before you can find perimeter or area — don't skip straight to adding up the labelled numbers only.
• Double-counting or missing a side when finding the perimeter of a compound shape. Trace around the shape systematically (e.g. clockwise) and tick off each side as you add it.
• Ignoring "whole tins/rolls only" conditions in problem-solving questions — if you can only buy whole units, round UP even if the exact calculation gives a decimal.
• Leaving out working. Even a partial, incorrect final answer can pick up method marks if your working (formulas, substitutions) is shown clearly.