Every measurement means something — but only if you use the right units and substitute into formulas correctly.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
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?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Come back to this at the end of the lesson.
Wrong: Converting centimetres to metres means dividing by 10.
Right: There are 100 centimetres in a metre, so divide by 100. Area conversions require dividing by 100² = 10 000.
Core Content
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:
A length of "5" is meaningless. Is it 5 mm? 5 km? The unit tells you the scale of the measurement. In every HSC answer, write the unit. A number without a unit scores no marks for the final answer.
Every metric length unit is a power of 10 away from its neighbour. Knowing the ladder lets you convert in one step.
| Converting from… | …to… | Operation | Example |
|---|---|---|---|
| mm | cm | ÷ 10 | 450 mm ÷ 10 = 45 cm |
| cm | m | ÷ 100 | 350 cm ÷ 100 = 3.5 m |
| m | km | ÷ 1000 | 2500 m ÷ 1000 = 2.5 km |
| km | m | × 1000 | 4.2 km × 1000 = 4200 m |
| m | cm | × 100 | 1.8 m × 100 = 180 cm |
| cm | mm | × 10 | 6.5 cm × 10 = 65 mm |
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 is length × length × length — so conversion factors are cubed. And capacity is just volume measured in litres.
$$1\text{ cm}^3 = 1000\text{ mm}^3 \qquad 1\text{ m}^3 = 1\,000\,000\text{ cm}^3$$Worked Examples
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 fence post is 1850 mm tall. Express this height in:
A bathroom tile has an area of 400 cm². Convert this to:
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
10 questions — work through these before checking answers.
Show all working. Write units in every answer.
1 Convert 3.6 km to metres.
2 Convert 850 mm to centimetres.
3 A rectangle is 12 m long and 7 m wide. Use $A = \ell \times w$ to find the area.
4 Convert 5.2 m² to cm².
5 Convert 3 500 000 mm² to m².
6 The formula for the area of a triangle is $A = \frac{1}{2}bh$. Find the area when $b = 14$ cm and $h = 9$ cm.
7 A box is 50 cm long, 20 cm wide and 15 cm high. Calculate its volume in cm³.
8 Convert the volume from Question 7 to litres.
9 A garden is 8.5 m long and 6 m wide. Find its area in: (a) m² (b) cm².
10 A water tank holds 2.5 kL. Express this in: (a) litres (b) cm³ (c) mL.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
Multiple Choice
1 A rectangle has a length of 3.5 m and a width of 80 cm. Which of the following correctly calculates the area in m²?
? 1 A rectangle has a length of 3.5 m and a width of 80 cm. Select the option that correctly calculates the area in m²?
2 How many mm² are there in 1 m²?
? Regarding this topic, 2 How many mm² are there in 1 m²?
3 A swimming pool has a volume of 48 000 L. What is this in kL?
? Regarding this topic, 3 A swimming pool has a volume of 48 000 L. What is this in kL?
Short Answer
SA 1 3 marks 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. (1 mark)
(b) Convert the area to cm². (1 mark)
(c) Convert the area to mm², expressing your answer in scientific notation. (1 mark)
SA 2 3 marks 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. (1 mark)
(b) Convert the volume to litres. (1 mark)
(c) Water is sold at $2.30 per kilolitre. Find the cost to fill the pool. (1 mark)
SA 3 2 marks A student calculates the area of a rectangle as follows:
Identify the error and write the correct solution. (2 marks)
SA 4 2 marks A paddock has an area of 3.6 hectares. Express this area in:
(a) m² (1 mark)
(b) km² (1 mark)
Climb platforms using your knowledge of formulas and unit conversions. Pool: lesson 1.
Tick when you have finished the lesson and checked your answers.