Units of measure

The Mars orbiter that cost 125 million dollars

• In 1999, NASA lost the Mars Climate Orbiter because one team used metres and another used feet.
• Unit mistakes aren't just embarrassing — they can be catastrophic. Getting units right is a core mathematical skill.

Metric 公制 length 长度, mass 质量 and capacity 容量

• Length: $10\text{ mm}=1\text{ cm}$, $100\text{ cm}=1\text{ m}$, $1000\text{ m}=1\text{ km}$. Moving to a larger unit divides the number.
• Mass: $1000\text{ g}=1\text{ kg}$. Divide by 1000 from g to kg; multiply from kg to g.
• Capacity: $1000\text{ ml}=1\text{ litre}$; $\;1\text{ litre} = 1000\text{ cm}^3$.
• $1\text{ m} = 100\text{ cm} = 1000\text{ mm}$.

Changing length units: multiply going down, divide going up

Area and volume conversions need care

• Area conversions are squared: $1\text{ m}^2 = 100^2 = 10\,000\text{ cm}^2$.
• Volume conversions are cubed: $1\text{ m}^3 = 100^3 = 1\,000\,000\text{ cm}^3$.

Worked examples

• $3\text{ m}^2 = 3 \times 10\,000 = 30\,000\text{ cm}^2$.
• $0.5\text{ m}^3 = 0.5 \times 1\,000\,000 = 500\,000\text{ cm}^3$.
• $2\,500\text{ cm}^3 = 2.5\text{ litres}$ (divide by $1000$).

Fun fact

• The metre was originally defined as $\dfrac{1}{10\,000\,000}$ of the distance from the equator to the North Pole. It's now defined using the speed of light: $1\text{ m} =$ the distance light travels in $\dfrac{1}{299\,792\,458}$ seconds.

Converting units is about moving between scales on this number line — from millimetres to kilometres and beyond.

You've got it

Write from 40000 square centimetres

• Convert 4 square metres to square centimetres.
• 1 square metre is 10 000 square centimetres.