Area - Measuring Two-Dimensional Space
Area is the amount of two-dimensional space inside a flat shape or surface. We use it every day – measuring floor space before buying tiles, calculating how much paint a wall needs, or finding the size of a garden.
Long before the hectare, English farmers measured land in acres – originally defined as roughly the area a team of oxen could plough in a single day, which is why an acre is a long, narrow shape rather than a neat square. That very practical, work-based definition survived for centuries and is still used for property and farmland in the UK and USA today, sitting alongside the metric hectare (about 2.47 acres) that most of the rest of the world adopted. Area measurement has high financial stakes even now: property is typically priced by area, so professional surveyors are legally required to measure floor area to strict, standardised rules, since even a small measurement dispute on a large building can represent a significant sum of money.
What Is Area?
Area is measured in square units. If you cover a shape with unit squares, the number of squares needed is the area. The unit must match the linear units used: centimetres give cm², metres give m².
Units of Area
| Unit | Symbol | Relation | Typical use |
|---|---|---|---|
| Square millimetre | mm² | 100 mm² = 1 cm² | Very small surfaces |
| Square centimetre | cm² | 10,000 cm² = 1 m² | Paper, small tiles |
| Square metre | m² | Base unit | Rooms, floor area |
| Hectare | ha | 1 ha = 10,000 m² | Agricultural land, parks |
| Square kilometre | km² | 1 km² = 100 ha | Countries, large regions |
Area Formulas
| Shape | Formula | Variables |
|---|---|---|
| Rectangle | A = l × w | l = length, w = width |
| Square | A = s² | s = side length |
| Triangle | A = ½ × b × h | b = base, h = perpendicular height |
| Circle | A = πr² | r = radius |
| Parallelogram | A = b × h | b = base, h = perpendicular height |
| Trapezium | A = ½(a + b) × h | a, b = parallel sides, h = height |
Worked Examples
A = 6 × 4.5 = 27 m².
A = ½ × 12 × 7 = 42 m².
3.5 × 10,000 = 35,000 m².
A = π(4)² = 16π ≈ 50.27 m².
Key Takeaways
- Area is measured in square units: mm², cm², m², ha, km².
- 1 m² = 10,000 cm² | 1 ha = 10,000 m² | 1 km² = 100 ha.
- Always use the perpendicular height in area formulas, not the slant height.