Compound Area
Section: Geometrical Reasoning, Shapes and Measurements | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Very Small and Very Large Units
Some quantities are measured using very small or very large units, depending on their size. Recognising the right unit - and converting between units - is an important measurement skill.
| Quantity | Very Small Unit | Very Large Unit |
|---|---|---|
| Length | millimetre (mm) | kilometre (km) |
| Mass | milligram (mg) | tonne (1000 kg) |
| Capacity | millilitre (ml) | megalitre (1,000,000 l) |
-
Question: A machine part is 4.2 mm thick. What is this in metres?
- Step 1: 1 mm = 0.001 m
- Step 2: 4.2 × 0.001 = 0.0042
- Answer: 0.0042 m
-
Question: A reservoir holds 3.6 megalitres of water. How many litres is this?
- Step 1: 1 megalitre = 1,000,000 litres
- Step 2: 3.6 × 1,000,000
- Answer: 3,600,000 litres
The Full Prefix System
Every metric prefix - for length, mass, OR capacity - stands for the same power of 10. Learning this one system lets you convert between ANY pair of prefixes, not just the common ones.
| Prefix | Letter | Multiply by | Power of 10 |
|---|---|---|---|
| tera | T | 1,000,000,000,000 | 1012 |
| giga | G | 1,000,000,000 | 109 |
| mega | M | 1,000,000 | 106 |
| kilo | k | 1,000 | 103 |
| (none) | - | 1 | 100 |
| centi | c | 0.01 | 10-2 |
| milli | m | 0.001 | 10-3 |
| micro | μ | 0.000001 | 10-6 |
| nano | n | 0.000000001 | 10-9 |
-
Question: A single grain of pollen has a mass of about 480,000,000 nanograms. Convert this to grams.
- Step 1: 1 nanogram = 0.000000001 g = 1 × 10⁻⁹ g
- Step 2: Instead of multiplying by 10⁻⁹, it's easier to divide by 1,000,000,000: 480,000,000 ÷ 1,000,000,000
- Answer: 0.48 g
-
Question: A data cable can transfer 2.5 gigabytes of data. Write this in bytes, then in megabytes.
- Step 1 (bytes): 1 gigabyte = 1,000,000,000 bytes, so 2.5 × 1,000,000,000
- Step 2 (megabytes): 1 gigabyte = 1000 megabytes, so 2.5 × 1000
- Answer: 2,500,000,000 bytes = 2500 megabytes
Area of Basic Shapes
Compound shapes are built from simple shapes - recap the key area formulae before combining them.
- Rectangle: Area = length × width
- Triangle: Area = ½ × base × height
- Circle: Area = πr²
Compound Area: Adding Shapes
Split a compound shape into simple shapes you recognise, find each area separately, then add them together.
An L-shape split into two rectangles with a dashed line
-
Question: An L-shaped garden is made of a rectangle 8 m by 3 m, joined to a rectangle 5 m by 4 m. Find the total area.
- Step 1: Area of rectangle 1 = 8 × 3 = 24 m²
- Step 2: Area of rectangle 2 = 5 × 4 = 20 m²
- Step 3: Add the areas: 24 + 20
- Answer: 44 m²
Compound Area: Subtracting Shapes
When a piece is cut out of a shape, find the area of the whole shape, then subtract the area of the missing piece.
-
Question: A rectangular lawn 12 m by 9 m has a rectangular flower bed 4 m by 3 m cut from one corner. Find the remaining lawn area.
- Step 1: Area of whole lawn = 12 × 9 = 108 m²
- Step 2: Area of flower bed = 4 × 3 = 12 m²
- Step 3: Subtract: 108 − 12
- Answer: 96 m²
Compound Area with Circles
Compound shapes often combine rectangles or triangles with circles, semicircles, or quarter circles. A useful shortcut: two semicircles of equal radius always combine to make one full circle.
-
Question: A running track has a straight rectangular section 40 m by 8 m, with a semicircular end of radius 4 m at each end. Find the total area (use π ≈ 3.14).
- Step 1: Rectangle area = 40 × 8 = 320 m²
- Step 2: The two semicircular ends combine to make one full circle: πr² = 3.14 × 4² = 50.24 m²
- Step 3: Add: 320 + 50.24
- Answer: 370.24 m²
-
Question: A square tile of side 10 cm has a circular hole of radius 2 cm cut from its centre. Find the remaining area (use π ≈ 3.14).
- Step 1: Square area = 10 × 10 = 100 cm²
- Step 2: Circle area = πr² = 3.14 × 2² = 12.56 cm²
- Step 3: Subtract: 100 − 12.56
- Answer: 87.44 cm²
Real-World Applications
Compound area calculations are used in many practical situations:
- Flooring and carpeting: estimating materials for L-shaped rooms.
- Running tracks: a rectangle with semicircular ends.
- Garden design: lawns with flower beds or paths cut out.
- Architectural plans: irregular floor plans built from simple shapes.
- Manufacturing: panels with circular holes cut for fittings.
Exam Tips
Common Mistakes
MistakeAdding all areas together even when a piece has been cut out
Fixfor shapes with a piece removed, subtract the smaller area from the larger one
MistakeUsing mismatched side lengths when splitting an L-shape into rectangles
Fixlabel every side length carefully before splitting, and check opposite sides add up correctly
MistakeDoubling the area of a full circle when two semicircular ends are shown
Fixtwo semicircles of equal radius combine to make exactly one full circle, not two
MistakeChoosing an unrealistic unit for a quantity, e.g. giving a car's mass in milligrams
Fixchoose a sensible unit for the size of the quantity - small units for tiny amounts, large units for huge amounts
MistakeMixing units within the same calculation, e.g. adding a length in cm to one in m without converting
Fixalways convert all measurements to the same unit before calculating
MistakeOnly knowing common prefixes like kilo and milli, and getting stuck on less familiar ones like nano or giga
Fixevery prefix stands for a power of 10 - learn the pattern once (tera to nano) and it applies to length, mass, or capacity
For Exams
Interactive revision notes, videos and practice questions load below.