Area & Perimeter
What are Area and Perimeter?
Perimeter is the total distance around the outside of a shape. It is a length, so it is measured in units such as cm, m, or km.
Area is the amount of space enclosed inside a shape. It is measured in square units such as cm², m², or km².
These two concepts are often confused. A key way to remember them: perimeter is like a fence around a field (one-dimensional distance), while area is like the grass inside the field (two-dimensional space).
Perimeter
To find the perimeter of any shape, add up all the side lengths.
Rectangle: P = 2(l + w) where l = length and w = width
- Rectangle 8cm × 5cm: P = 2(8 + 5) = 2 × 13 = 26cm
Square: P = 4s where s = side length
- Square with side 7cm: P = 4 × 7 = 28cm
Triangle: P = a + b + c (sum of all three sides)
- Sides 5cm, 7cm, 9cm: P = 5 + 7 + 9 = 21cm
For irregular polygons, measure and add every side individually.
Area of Rectangles and Squares
Rectangle: A = l × w
- Length 8cm, width 5cm: A = 8 × 5 = 40cm²
Square: A = s²
- Side 9cm: A = 9² = 81cm²
Note that the area of a rectangle (40cm²) is different from its perimeter (26cm) — always check which one the question is asking for.
Area of Triangles and Parallelograms
Triangle: A = 1/2 × base × perpendicular height
- Base 10cm, height 6cm: A = 1/2 × 10 × 6 = 30cm²
The height must be perpendicular (at 90°) to the base — not a slanted side.
Parallelogram: A = base × perpendicular height
- Base 12cm, height 5cm: A = 12 × 5 = 60cm²
A parallelogram has the same area formula as a rectangle when the perpendicular height is used.
Area of Trapezium
A trapezium has one pair of parallel sides (called a and b) and a perpendicular height h.
Trapezium: A = 1/2 × (a + b) × h
Worked example: Parallel sides 5cm and 9cm, height 4cm
- A = 1/2 × (5 + 9) × 4 = 1/2 × 14 × 4 = 28cm²
Think of it as finding the average of the two parallel sides, then multiplying by the height.
Circles: Circumference and Area
For circles, all formulas involve π (pi) ≈ 3.14159...
Circumference (perimeter of a circle): C = πd = 2πr
- Radius 7cm: C = 2 × π × 7 ≈ 2 × 3.14 × 7 = 43.96cm
Area of a circle: A = πr²
- Radius 7cm: A = π × 7² = π × 49 ≈ 3.14 × 49 = 153.86cm²
Remember: circumference uses radius or diameter; area always uses radius squared.
Key Terms
| Term | Meaning |
|---|---|
| Perimeter | Total length around the outside of a shape |
| Area | Space enclosed inside a shape |
| Perpendicular height | Height measured at 90° to the base |
| Radius (r) | Distance from centre to edge of a circle |
| Diameter (d) | Distance across a circle through the centre (d = 2r) |
| π (pi) | Approximately 3.14159, the ratio of circumference to diameter |
Composite Shapes
A composite shape is made by joining two or more simple shapes together. To find the area:
- Split the composite shape into rectangles, triangles, and/or semicircles.
- Calculate the area of each part.
- Add (or subtract) the parts.
Example: An L-shaped room can be split into two rectangles. Find each rectangle's area, then add them together.
If a shape has a region removed (e.g. a square with a circular hole), calculate the area of the whole shape then subtract the area of the removed part.
Units and Conversions
Always use consistent units throughout a calculation.
| Conversion | Equivalent |
|---|---|
| 1 m = 100 cm | 1 m² = 10,000 cm² |
| 1 km = 1,000 m | 1 km² = 1,000,000 m² |
Note that unit conversions for area require squaring the linear conversion factor — a common source of error.
Common Mistakes
- Confusing area and perimeter — re-read the question carefully and check what unit the answer should be in.
- Using slant height instead of perpendicular height — always use the height that is at 90° to the base.
- Forgetting to square r in the circle area formula — A = πr², not πr.
- Using diameter instead of radius — check which measurement you have been given before applying a formula.
- Not converting units — mixing cm and m in the same calculation gives a wrong answer.
Tips and Tricks
- Memorise the key formulas as a group: rectangle (l × w), triangle (1/2 × b × h), circle area (πr²), circumference (2πr).
- For composite shapes, draw dotted lines to show how you are splitting them — this prevents double-counting.
- A quick check: area answers must always have squared units (cm², m²); perimeter answers have plain units (cm, m).
- If you know the perimeter of a square, divide by 4 to get the side, then square for the area — a two-step shortcut.