Area and Perimeter
Perimeter is the distance around a shape; area is the space inside it. Two ideas cause almost every lost mark: confusing the two, and using a slanted side instead of the perpendicular height.
1. The formulae
| Shape | Area |
|---|---|
| Rectangle | length × width |
| Triangle | ½ × base × perpendicular height |
| Parallelogram | base × perpendicular height |
| Trapezium | ½(a + b) × h, where a and b are the parallel sides |
| Kite / rhombus | ½ × d₁ × d₂ (the diagonals) |
| Circle | πr² |
Check which of these your formula list provides. Tutors said some of these must be memorised while others appear on the sheet — confirm against your own exam series so you know what to learn.
Perimeter is always just the sum of all the outside lengths. There is no formula to remember — but every side must be included.
Area and perimeter are different measurements, and the units prove it. Perimeter is a length (cm); area is cm². If your answer has the wrong unit, you’ve calculated the wrong thing — the fastest check available.
2. Perpendicular height — the big one
“Height” always means the PERPENDICULAR height — measured at 90° to the base, not along a sloping side.
For a triangle or parallelogram, diagrams deliberately show both the slant length and the perpendicular height, and you must pick the right one.
The slanted side is not the height. Using it was the most common error recorded in this topic, and it always makes the area too large.
For a trapezium, h is the distance between the two parallel sides, again perpendicular.
a and b must be the PARALLEL sides. Picking the wrong pair — the sloping ones — was specifically recorded. Identify the parallel sides first (they’re usually marked with arrows), then measure between them.
Order of operations in the trapezium formula: add the parallel sides, halve, then multiply by the height. Writing it as (a + b)/2 × h makes the order clear, and doing the multiplication before the addition was a documented error.
3. Compound shapes
Shapes made from rectangles, triangles and semicircles joined together.
For AREA:
- Split the shape into parts you recognise — draw the dividing lines on the diagram
- Find any missing lengths by subtracting
- Calculate each part
- Add them (or subtract, for a cut-out)
Split it, don’t guess. Breaking a pentagon or an L-shape into a rectangle plus a triangle was where students got stuck most often. Draw the line and label every piece.
Finding missing lengths: in an L-shape, opposite sides of the overall rectangle must be equal, so a missing length is the difference between two given ones.
For PERIMETER:
Trace around the outside and add every edge you cross. Do not include the internal lines you drew to split the shape.
Internal dividing lines are not part of the perimeter. This is the classic compound-shape error — the splitting lines helped you find the area, but they are inside the shape.
4. Shaded regions
Shaded area = whole shape − unshaded part
Example: a circle of radius 5 cm inside a square of side 12 cm. Shaded area outside the circle:
- Square = 144 cm²
- Circle = π(25) = 78.54 cm²
- Shaded = 65.5 cm² (3 s.f.)
Decide what to subtract from what before you start, and label both parts on your sketch. A negative answer means you subtracted the wrong way round.
Don’t round before subtracting — carry full accuracy through.
5. Working backwards
Given the area and asked for a missing length, substitute and rearrange.
Example: a triangle has area 24 cm² and base 8 cm. Find its height.
- 24 = ½ × 8 × h → 24 = 4h → h = 6 cm
Example: a rectangle has perimeter 30 cm and length 9 cm.
- 2(9 + w) = 30 → 9 + w = 15 → w = 6 cm
6. Units
Convert to the same unit BEFORE calculating, and remember that area conversions square the factor.
| Conversion | Factor |
|---|---|
| 1 m = 100 cm | ×100 |
| 1 m² = 10 000 cm² | ×100² |
| 1 cm = 10 mm | ×10 |
| 1 cm² = 100 mm² | ×10² |
Always state units, squared for areas. Marks are given for them.
7. Mistakes that cost marks
Confusing area with perimeter.
Using a slanted side as the height.
Choosing the wrong pair of sides as the parallel ones in a trapezium.
Getting the order wrong in the trapezium formula.
Forgetting the ½ for a triangle.
Including internal dividing lines in a perimeter.
Missing a side when adding up a perimeter.
Subtracting the wrong way round in a shaded region.
Using a linear factor to convert area units.
Omitting units, or not squaring them.
Frequently asked questions
What is the difference between area and perimeter? Perimeter is the distance around (a length, cm). Area is the space inside (cm²).
What is the area of a triangle? ½ × base × perpendicular height.
What is the area of a trapezium? ½(a + b) × h, where a and b are the parallel sides and h is the perpendicular distance between them.
Which height do I use? Always the perpendicular height — never a sloping side.
How do I find the area of a compound shape? Split it into familiar shapes, find any missing lengths, and add or subtract the areas.
How do I find the perimeter of a compound shape? Add every outside edge — excluding the internal lines you used to split it.
How do I find a shaded area? Whole shape minus the unshaded part.
How many cm² are in a m²? 10 000 — the factor is squared.
Do I need to memorise the formulae? Some are given and some are not — check your own formula list.
Quick revision checklist
- I know the area formulae for rectangle, triangle, parallelogram and trapezium
- I know which formulae my exam provides
- I always use the perpendicular height
- I identify the parallel sides in a trapezium
- I apply the trapezium formula in the right order
- I can split compound shapes and find missing lengths
- I exclude internal lines from a perimeter
- I check every outside edge is included
- I can find shaded areas by subtraction
- I keep full accuracy before subtracting
- I can work backwards from an area or perimeter
- I convert area units by squaring the factor
- I give units, squared for area
These notes cover area and perimeter in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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