Year 10 Standard · Unit 1 Number sets, surds and accuracy · Lesson 3 of 6

How many square centimetres fit in one square metre?

Metric prefixes are powers of ten, from nano to tera, so changing a length is multiplying or dividing by a power of ten. Areas and volumes need more care: draw the square or the cube, and count the small ones inside.

Number and AlgebraGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: write 3.2 × 10⁵ as an ordinary number.
    Answer320 000. The digits move 5 places to the left.
  2. From Year 8: how many centimetres are there in 4.5 m?
    Answer450 cm. There are 100 cm in 1 m, and 4.5 × 100 = 450.
  3. From Year 9: work out 10³ × 10⁴, as a power of 10.
    Answer10⁷. Multiplying powers of the same base adds the indices: 3 + 4 = 7.
Words used in this lesson
prefix
a part at the start of a unit that multiplies it by a power of ten. For example: kilo means × 1000.
square centimetre
the area of a square 1 cm by 1 cm, written cm². For example: a postage stamp is about 6 cm².
cubic metre
the volume of a cube 1 m along each edge, written m³. For example: a large packing crate is about 1 m³.
litre
a unit of volume equal to 1000 cm³. For example: a 1.5 litre bottle.
density
mass for each unit of volume. For example: gold: 19.3 g/cm³.
speed
distance for each unit of time. For example: 30 m/s.
conversion factor
the number you multiply or divide by to change units. For example: 1000 for km to m.

Ordering turf

A garden centre in Cape Town sells turf by the square metre, but the plan for a courtyard is drawn in centimetres: it is 350 cm by 400 cm.

Predict first: how many square centimetres are there in one square metre, 100 or 10 000? So is the courtyard 14 m², 140 m² or 1400 m²? Then use the first figure to check.

m to cm
1 m = 100 cmone big unit is
× 100to convert, multiply by

1 m = 100 cm, so to change m to cm, multiply by 100.

Show me the steps
  1. Length, metres to centimetres: 1 m = 100 cm. Each of the 10 parts is 10 cm.
  2. Press Area: 1 m² is a square 100 cm by 100 cm. Each small square is 10 cm by 10 cm = 100 cm², and there are 100 of them, so 1 m² = 100 × 100 = 10 000 cm².
  3. Press Volume: 1 m³ is a cube 100 cm along each edge, so 1 m³ = 100 × 100 × 100 = 1 000 000 cm³.
  4. Slide to km to m and press Area: 1 km² = 1000 × 1000 = 1 000 000 m². The length factor, 1000, is used twice.

A length factor is used once for a length, twice for an area and three times for a volume. 1 m = 100 cm, so 1 m² = 100 × 100 = 10 000 cm² and 1 m³ = 100 × 100 × 100 = 1 000 000 cm³. A litre is 1000 cm³, so 1 m³ = 1000 litres, and 1 cm³ = 1 mL.

Prefixes are powers of ten: tera (T) 10¹², giga (G) 10⁹, mega (M) 10⁶, kilo (k) 10³, centi (c) 10⁻², milli (m) 10⁻³, micro (μ) 10⁻⁶ and nano (n) 10⁻⁹. Changing to a smaller unit gives more of them, so multiply; changing to a bigger unit gives fewer, so divide. A compound unit such as m/s or g/cm³ is converted one part at a time.

km
mm
1 km = 10⁶ mmone unit is
2 down, × 1000 eachsteps on the ladder

From km to mm is 2 steps down the ladder, each × 1000, so 1 km = 10⁶ mm = 1 000 000 mm.

Show me the steps
  1. From km to mm is 2 steps down the ladder (kilo to plain metres, then to milli), each × 1000, so 1 km = 10⁶ mm = 1 000 000 mm.
  2. Set From to Gm and To to km: 2 steps down, so 1 Gm = 10⁶ km. The same two steps give the same factor anywhere on the ladder.
  3. Set From to nm and To to m: 3 steps up, each ÷ 1000, so 1 nm = 10⁻⁹ m. Going up to a bigger unit, there are fewer of them.
  4. Set From and To to the same rung: no steps, so the number does not change.

A common mistake: using the length factor for an area or a volume, as in “1 m² = 100 cm²”. Draw the square: it is 100 cm along each side, so it holds 100 rows of 100 squares. The other slip is multiplying when going to a bigger unit; check that the number of big units is smaller.

Worked examples

1. Convert 0.35 m² to cm².

  1. It is an area, so the factor is 100 × 100 = 10 000.
  2. Square centimetres are smaller, so there are more of them: 0.35 × 10 000 = 3500 cm².

2. A red blood cell is 7.5 μm across. Write this in mm.

  1. Micro to milli is one step up the ladder, from 10⁻⁶ to 10⁻³, so divide by 1000.
  2. 7.5 ÷ 1000 = 0.0075 mm. A millimetre is bigger, so there are fewer of them.

3. A cyclist rides at 20 m/s. Write this speed in km/h.

  1. Change one part of the unit at a time. In one hour (3600 s) the cyclist goes 20 × 3600 = 72 000 m.
  2. 72 000 m = 72 km, so 20 m/s = 72 km/h.

Your go. The first step is given. A fish tank holds 0.0042 m³ of water. How many litres is that?

  1. 1 m³ = 1 000 000 cm³, so 0.0042 m³ = 0.0042 × 1 000 000 = 4200 cm³.
  2. Now use 1 litre = 1000 cm³.
Check the answer

4200 ÷ 1000 = 4.2 litres.

With a GDC (optional)

Type powers of ten with the ×10ˣ or EE key, not with × 10 and the power key. The distance from the Sun to the Earth is about 1.5 × 10¹¹ m and light travels 3 × 10⁸ m every second, so type 1.5 EE 11 ÷ 3 EE 8: the GDC shows 500, the number of seconds the light takes.

A GDC does not know which unit a number is in. Decide the factor first (is it 100, 100², 100³ or a step of 1000?), then use the GDC for the arithmetic.

Check before you go on

1. How many square centimetres are there in 4 m²?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A virus particle is 120 nm across. What is that in micrometres (μm)?

3. A solar farm in the Atacama Desert produces 1.4 GW at its peak. What is that in megawatts (MW)?

Level 2 · Building

4. A solar panel on a roof in Rajasthan has an area of 1.8 m². What is its area in cm²?

5. A rainwater tank in Brisbane holds 0.75 m³. How many litres is that? (1 litre = 1000 cm³.)

6. A cheetah in Namibia sprints at 30 m/s. What is this speed in km/h?

Level 3 · Stretch

7. A memory drive holds 2 TB. Each photo takes up 4 MB. Using 1 TB = 10¹² bytes and 1 MB = 10⁶ bytes, how many photos fit on the drive?

8. Gold has a density of 19.3 g/cm³. What is this in kg/m³?

Something to think about

On a map with scale 1 : 50 000, a lake covers 3 cm². What is the real area of the lake in km²?

One way to think about it

1 cm on the map is 50 000 cm = 500 m = 0.5 km on the ground. So 1 cm² on the map is 0.5 km × 0.5 km = 0.25 km², because the scale factor is used twice for an area. The lake is 3 × 0.25 = 0.75 km². Using 0.5 only once would give 1.5 km², double the true area.

Exit ticket

9. A medicine syringe holds 2.5 mL. How many cubic millimetres (mm³) is that? (1 mL = 1 cm³.)