8.1 Length & Perimeter
Key Takeaways
- The metric ladder is 10 mm = 1 cm, 100 cm = 1 m and 1000 m = 1 km, so multiplying moves you to smaller units and dividing moves you to larger ones
- Choose the unit that gives a sensible number: mm for tiny objects, cm or m for rooms and people, km for distances between places
- Perimeter is the total distance around the outside of a shape; add every side, including sides you must work out first
- In rectilinear composite shapes, missing horizontal sides add up to the total length and missing vertical sides add up to the total height
- Circumference C = pi x d (or 2 x pi x r); with pi approx 3.14, a circle of diameter 10 cm has circumference about 31.4 cm
Measures & Units is one of the five ICAS Mathematics strands, and length is its foundation. From Paper A (Year 3), where you might be asked which unit suits a pencil, through to Papers G-H (Years 9-10), where geometrical measurement appears inside multi-step problems, you will always need fluent metric conversions and reliable perimeter methods. Since personal calculators are not allowed and an on-screen calculator only appears when a question truly needs one, you should be able to convert units and add strings of lengths mentally or with quick written working.
Metric Units of Length
The metric system is built on powers of ten, which is why conversions are really just place-value shifts.
| Unit | Symbol | Relationship | Typical use |
|---|---|---|---|
| millimetre | mm | 10 mm = 1 cm | thickness of a coin, width of a pencil lead |
| centimetre | cm | 100 cm = 1 m | length of a pencil, height of a mug |
| metre | m | 1000 m = 1 km | height of a door, length of a classroom |
| kilometre | km | 1 km = 1000 m | distance between suburbs |
Choosing a Sensible Unit
ICAS loves "which unit would you use to measure..." questions. The trick is to pick the unit that gives a convenient number, usually between about 1 and 100. A cricket pitch is about 20 m (not 2000 cm); an ant is about 5 mm (not 0.005 m). If your answer would be a very large or very small number, you probably chose the wrong unit.
Converting Between Units
- Big to small unit: multiply. 3.4 m = 3.4 x 100 = 340 cm.
- Small to big unit: divide. 2500 m = 2500 / 1000 = 2.5 km.
- Chained conversions: 2 km = 2000 m = 200,000 cm.
A common ICAS trap is mixing units inside one question: "A ribbon is 2 m long. Mia cuts off 35 cm. How much is left?" Convert first (200 cm - 35 cm = 165 cm), never subtract across different units.
Measuring and Estimating
Younger papers ask you to read a ruler or estimate. Read from zero, not from the end of the ruler, and watch for scales marked in millimetres. For estimation, anchor to known benchmarks: a door is about 2 m, a fingernail about 1 cm, a school oval lap about 400 m. Estimate first, then measure — ICAS questions often ask which estimate is most reasonable.
Perimeter
Perimeter is the total distance around the outside of a shape. Add the lengths of all the sides.
- Square: P = 4 x side. Side 7 cm gives P = 28 cm.
- Rectangle: P = 2 x (length + width). 9 cm by 4 cm gives P = 2 x 13 = 26 cm.
- Any polygon: add every side, one by one, and check you have counted each side exactly once.
Worked Example 1 (Papers B-C style)
A rectangular garden bed is 6 m long and 3.5 m wide. How much edging is needed to go all the way around?
P = 2 x (6 + 3.5) = 2 x 9.5 = 19 m. Estimation check: about 2 x 10 = 20 m, so 19 m is sensible.
Composite Shapes with Missing Sides
From Papers C-D onward, ICAS shows an L-shaped or stepped figure with only some sides labelled and asks for the perimeter. The key facts:
- All the missing horizontal pieces together equal the total length of the shape.
- All the missing vertical pieces together equal the total height.
Worked Example 2
An L-shape fits inside a 10 cm by 8 cm bounding rectangle. Two inner sides are labelled 4 cm and 3 cm. Find the perimeter.
For any rectilinear shape made by cutting a rectangle corner out, the perimeter equals that of the bounding rectangle: P = 2 x (10 + 8) = 36 cm. The 4 cm and 3 cm inner sides are red herrings for the perimeter (they would matter for area). Verify: the missing horizontal side is 10 - 4 = 6 cm and the missing vertical side is 8 - 3 = 5 cm; adding all six sides gives 10 + 8 + 6 + 5 + 4 + 3 = 36 cm.
Working Backwards
ICAS also asks reverse questions: "A rectangle has perimeter 30 cm and length 9 cm. Find its width." Half the perimeter is length + width, so 15 - 9 = 6 cm.
Circumference of a Circle (Older Papers)
From around Papers E-F, circles appear. The circumference is the perimeter of a circle:
C = pi x d = 2 x pi x r
where d is the diameter and r is the radius. Using pi approx 3.14, a circle with radius 5 cm has C = 2 x 3.14 x 5 = 31.4 cm. Without a calculator, use 3.14 or the fraction 22/7 when the radius is a multiple of 7. Remember that diameter is twice the radius — a classic trap is plugging the diameter into 2 x pi x r.
Worked Example 3
A circular running track has diameter 70 m. Using pi approx 22/7, how far is one lap?
C = pi x d = 22/7 x 70 = 220 m. Choosing 22/7 makes the sevens cancel, which is exactly the sort of mental arithmetic ICAS rewards.
Word Problem Strategy
- Underline what is asked and note every unit.
- Convert all measurements to the same unit before calculating.
- Compute, then estimate to check the size of your answer.
- Re-read the question: does it want cm, m, or km in the answer?
A skipping rope is 2.4 m long. What is its length in centimetres?
A rectangle has a perimeter of 36 cm. If its length is 11 cm, what is its width?