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 calculatingwrite 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.
Formulas use symbols to connect quantities; substitute in a 5-step method, write formula, check units match, substitute values, evaluate, write answer with unit. A number without a unit scores zero in the HSC.
Pause, copy the 5-step substitution method (write formula → check units → substitute → evaluate → state answer with unit) and the length unit ladder (mm ÷10 cm ÷100 m ÷1000 km) into your book.
Quick check: To convert 450 mm to centimetres, which operation do you apply?
We just saw the 5-step substitution method and the length unit ladder, where each step is a power of 10. That raises a question: if 1 m = 100 cm, does that mean 1 m² = 100 cm²? This card answers it → area conversions square the length factor (1 m² = 10 000 cm²) and volume conversions cube it (1 m³ = 1 000 000 cm³), because area is length × length and volume is length × length × length.
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
Area conversions square the length factor (1 m² = 10 000 cm²); volume conversions cube it (1 m³ = 1 000 000 cm³). Capacity bridge: 1 cm³ = 1 mL exactly; 1 m³ = 1 kL. 1 ha = 10 000 m².
Pause, copy the area squaring rule (1 m² = 10 000 cm²), the volume cubing rule (1 m³ = 1 000 000 cm³), and the capacity bridge (1 cm³ = 1 mL; 1 m³ = 1 kL) into your book.
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.
$= 60 \times 30 \times 40$
$= 72\,000\text{ cm}^3$
$72\,000\text{ cm}^3 = 72\,000\text{ mL}$
$= 72\,000 \div 1000 = 72\text{ L}$
$72 \div 1000 = 0.072\text{ kL}$
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
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 confidencethat 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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