Working With Formulas and Units
Every measurement means something — but only if you use the right units and substitute into formulas correctly. This lesson builds the foundation: unit ladders, area and volume conversions, and the four-step substitution method that earns full marks every time.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
A recipe calls for 250 mL of milk, but you only have a tablespoon (15 mL). How many tablespoons do you need? What did you have to do before you could answer — and how is that like what we do in maths?
Without calculating — write your gut feeling. We'll revisit this at the end of the lesson.
Measurement in Maths Standard starts with two core formulas and a system of unit conversions. Lock these in — every other measurement topic builds from them.
Area is length × length, so area units are always squared. Volume is length × length × length, so volume units are cubed. When you convert a length unit, area factors square and volume factors cube.
Key facts
- The metric units for length, area, volume and capacity
- Conversion factors between mm, cm, m and km
- Conversion factors for area and volume units
- What it means to substitute into a formula
- The abbreviations ha (hectare) and kL (kilolitre)
Concepts
- Why area conversions square the length factor
- Why volume conversions cube the length factor
- Why units must match before substituting
- How to read a formula and identify what to find
Skills
- Substitute values into a given formula and evaluate
- Convert between length, area and volume units
- Convert between volume and capacity units
- Identify and correct unit-mismatch errors
A formula is a rule written in symbols that tells you how quantities are related — substitute numbers in, and it does the work for you.
In Maths Standard, formulas are always given. Your job is to:
- Identify which variable you need to find
- Check that all values are in consistent units
- Substitute the known values
- Evaluate (calculate) the answer
- Write the answer with the correct unit
Every metric length unit is a power of 10 away from its neighbour. Knowing the ladder lets you convert in one step.
What to write in your book
- The 4-step substitution method: write formula → check units → substitute → evaluate → answer with unit.
- Length ladder: mm ÷ 10 = cm ÷ 100 = m ÷ 1000 = km. Reverse the operation when going back.
- A number without a unit scores no marks for a final answer in the HSC.
Quick check: To convert 450 mm to centimetres, which operation do you apply?
Area is length × length — so when you convert the length unit, you must square the conversion factor too.
Think of a square that is 1 cm × 1 cm = 1 cm². How many mm² is that? Each side is 10 mm, so the area is 10 × 10 = 100 mm². The conversion factor for lengths (×10) becomes ×100 for areas.
$$1\text{ cm}^2 = 100\text{ mm}^2 \qquad 1\text{ m}^2 = 10\,000\text{ cm}^2 \qquad 1\text{ km}^2 = 1\,000\,000\text{ m}^2$$Volume — units get cubed. Volume is length × length × length — so conversion factors are cubed. And capacity is just volume measured in litres.
$$1\text{ cm}^3 = 1000\text{ mm}^3 \qquad 1\text{ m}^3 = 1\,000\,000\text{ cm}^3$$Volume ↔ Capacity bridge: 1 cm³ = 1 mL · 1 000 cm³ = 1 L · 1 m³ = 1 000 L = 1 kL
What to write in your book
- Area conversions: square the length factor. 1 m² = 10 000 cm² because 1 m = 100 cm and 100² = 10 000.
- Volume conversions: cube the length factor. 1 m³ = 1 000 000 cm³ because 100³ = 1 000 000.
- Capacity bridge: 1 cm³ = 1 mL (exact). 1 kL = 1000 L = 1 m³.
- 1 ha = 10 000 m². Between m² and km² on the area ladder.
True or false: To convert 5 m² to cm², you multiply by 100 (because 1 m = 100 cm).
Worked examples · 3 in a row, reveal as you go
The formula for the area of a trapezium is $A = \dfrac{1}{2}(a + b)h$. Find the area when $a = 6$ cm, $b = 10$ cm and $h = 4$ cm.
$A = \dfrac{1}{2}(a + b)h$
$A = \dfrac{1}{2}(6 + 10) \times 4$
$A = \dfrac{1}{2}(16) \times 4$
$A = 8 \times 4 = 32$
$A = 32\text{ cm}^2$
A fence post is 1850 mm tall. Express this height in (a) centimetres and (b) metres. Also: a bathroom tile has area 400 cm² — convert to (c) mm² and (d) m².
$1850 \div 10 = 185\text{ cm}$
$1850 \div 1000 = 1.85\text{ m}$
$400 \times 100 = 40\,000\text{ mm}^2$
$400 \div 10\,000 = 0.04\text{ m}^2$
A rectangular fish tank measures 60 cm long, 30 cm wide and 40 cm high. Find: (a) the volume in cm³, (b) the capacity in litres, (c) the capacity in kL.
$= 60 \times 30 \times 40$
$= 72\,000\text{ cm}^3$
$72\,000\text{ cm}^3 = 72\,000\text{ mL}$
$= 72\,000 \div 1000 = 72\text{ L}$
$72 \div 1000 = 0.072\text{ kL}$
What to write in your book
- Formula substitution: write formula → substitute → evaluate → answer with unit.
- Volume → capacity: 1 cm³ = 1 mL (exact). Divide by 1000 to go from mL to L, divide by 1000 again for kL.
- All three dimensions must be in the same unit before multiplying for volume.
Fill the gap: A box with volume 15 000 cm³ holds litres of water (because 1 cm³ = 1 mL, and 1000 mL = 1 L).
Common errors · the 3 traps that cost marks
What to write in your book
- Before substituting into any formula, check all values share the same length unit.
- Converting 5 m² to cm²: multiply by 100² = 10 000 → 50 000 cm². Not 500.
- Every numerical answer in the HSC requires a unit.
Match each conversion: Drag or select the correct factor.
Quick-fire practice · 5 calculations
Convert 3.6 km to metres.
Convert 850 mm to centimetres.
A rectangle is 12 m long and 7 m wide. Use $A = \ell \times w$ to find the area.
Convert 5.2 m² to cm².
A box is 50 cm long, 20 cm wide and 15 cm high. Calculate its volume in cm³, then convert to litres.
Top 3 list: Name THREE different units that can be used to measure area (not volume, not length).
Look back at what you wrote in the Think First section. For the tablespoon problem: 250 ÷ 15 ≈ 16.7 tablespoons. You had to convert the question into a division — which is exactly what formula substitution does: turns a word problem into a calculation.
What has changed? What did you get right? What surprised you?
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
SA 1. The formula for the area of a circle is $A = \pi r^2$. A circular fountain has a radius of 2.5 m. (a) Find the area of the fountain in m², correct to 2 decimal places. (b) Convert the area to cm². (c) Convert the area to mm², expressing your answer in scientific notation. (3 marks)
SA 2. A rectangular swimming pool is 25 m long, 10 m wide and 1.8 m deep. (a) Calculate the volume of the pool in m³. (b) Convert the volume to litres. (c) Water is sold at $2.30 per kilolitre. Find the cost to fill the pool. (3 marks)
SA 3. A student calculates the area of a rectangle: Length = 4 m, Width = 50 cm. $A = 4 \times 50 = 200\text{ m}^2$. Identify the error and write the correct solution. (2 marks)
SA 4. A paddock has an area of 3.6 hectares. Express this area in: (a) m² and (b) km². (2 marks)
📖 Comprehensive answers (click to reveal)
Drill 1: 3.6 × 1000 = 3600 m · 2: 850 ÷ 10 = 85 cm · 3: $A = 12 \times 7 = $ 84 m² · 4: $5.2 \times 10\,000 = $ 52 000 cm² · 5: $V = 50 \times 20 \times 15 = 15\,000\text{ cm}^3 = $ 15 L
SA 1 (3 marks): (a) $A = \pi \times 2.5^2 = 19.63\text{ m}^2$ [1]. (b) $19.63 \times 10\,000 = 196\,350\text{ cm}^2$ [1]. (c) $196\,350 \times 100 = 1.9635 \times 10^7\text{ mm}^2$ [1].
SA 2 (3 marks): (a) $V = 25 \times 10 \times 1.8 = 450\text{ m}^3$ [1]. (b) $450 \times 1000 = 450\,000\text{ L}$ [1]. (c) $450\,000 \div 1000 = 450\text{ kL}$; cost = $450 \times 2.30 = \$1035$ [1].
SA 3 (2 marks): Error — width not converted to metres before substituting [1]. Correct: 50 cm = 0.5 m; $A = 4 \times 0.5 = 2\text{ m}^2$ [1].
SA 4 (2 marks): (a) $3.6 \times 10\,000 = 36\,000\text{ m}^2$ [1]. (b) $36\,000 \div 1\,000\,000 = 0.036\text{ km}^2$ [1].
Five timed questions on formulas and unit conversions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering questions on formulas and unit conversions. Pool: lesson 1.
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