What is area?
Area is how much flat surface a shape covers. It’s measured in square units: how many 1 cm × 1 cm squares (cm²) or 1 m × 1 m squares (m²) fit inside.
The formulas
| Shape | Formula | Why |
|---|---|---|
| Rectangle | A = l × w | rows × columns of squares |
| Square | A = s² | a rectangle with equal sides |
| Parallelogram | A = b × h | cut off a triangle and slide it to make a rectangle |
| Triangle | A = ½ × b × h | half of a parallelogram |
| Trapezoid | A = ½ × (a + b) × h | two copies make a parallelogram |
| Circle | A = πr² | (see the circle lessons) |
Base and height
The height must be perpendicular (at 90°) to the base. It is not the slanted side.
For a parallelogram with base 10 cm, slanted side 6 cm and height 5 cm:
A = 10 × 5 = 50 cm² (the 6 cm side isn’t used).
For an obtuse triangle, the height may fall outside the triangle; extend the base to meet it.
Worked examples for each shape
Rectangle: 8 m by 3.5 m → A = 28 m²
Triangle: base 12 cm, height 7 cm → A = ½ × 12 × 7 = 42 cm²
Trapezoid: parallel sides 6 m and 10 m, height 4 m → A = ½ × 16 × 4 = 32 m²
Circle: r = 3 cm → A = 9π ≈ 28.3 cm²
Working backwards
Example: A rectangle has area 54 cm² and width 6 cm. Length?
l = 54 ÷ 6 = 9 cm.
Example: A triangle has area 30 m² and base 12 m. Height?
30 = ½ × 12 × h → 30 = 6h → h = 5 m.
Units
- Area is always in square units: cm², m², km², ha.
- Make sure all lengths use the same unit before you multiply.
Area vs perimeter
| Perimeter | Area |
|---|---|
| distance around | surface inside |
| units: cm | units: cm² |
| fence | grass |
Two shapes can have the same perimeter but different areas. A 1 × 9 rectangle and a 5 × 5 square both have perimeter 20, but areas 9 and 25.
Common mistakes
- Using the slant side instead of the perpendicular height.
- Forgetting the ½ for triangles and trapezoids.
- Writing cm instead of cm².
- Mixing units (one side in m and one in cm).
Try it yourself
- Area of a square with side 9 cm
- Area of a parallelogram with base 15 m and height 4 m
- Area of a triangle with base 9 cm and height 6 cm
- Area of a trapezoid with parallel sides 5 and 11 and height 6
- A rectangle has area 72 m² and length 12 m. Width?
Answers: 1) 81 cm² 2) 60 m² 3) 27 cm² 4) 48 square units 5) 6 m
Quick recap
- Area measures surface in square units.
- Learn the formulas: rectangle, parallelogram, triangle, trapezoid, circle.
- Always use the perpendicular height.