POP

E5.1 Units of Measure

Image · 914 × 270

01 — Explanation

What are the standard metric units for length, mass and capacity?

Length: mm, cm, m, km
Mass: g, kg
Capacity: ml, l
Area: mm², cm², m², km²
Volume: mm³, cm³, m³

02 — Explanation

How do I convert between simple length, mass or capacity units?

Multiply or divide by the conversion factor, moving in the direction that matches the unit change: $\times 1000$ to go from km→m or kg→g or l→ml, and $\div 1000$ to reverse it; $\times 10$ / $\div 10$ between cm and mm; $\times 100$ / $\div 100$ between m and cm.

03 — Explanation

Why can't I use the same conversion factor for area units?

Because area is length × length, converting an area means applying the length conversion factor twice (squared):

1\text{ m}^2 = 100\text{ cm} \times 100\text{ cm} = 10\,000\text{ cm}^2 \quad \text{(not simply 100 cm}^2\text{)}

So converting cm² to m² means dividing by $100^2 = 10\,000$ , not by $100$ .

04 — Explanation

How does this extend to converting volume and capacity?

Volume is length × length × length, so the conversion factor is applied three times (cubed):

1\text{ m}^3 = 100^3\text{ cm}^3 = 1\,000\,000\text{ cm}^3

Volume and capacity are also directly linked: $1\text{ cm}^3 = 1\text{ ml}$ , and therefore $1\text{ m}^3 = 1000$ litres.

05 — Explanation

How do I handle a "compound" unit, like a rate given in g/cm³ or km/h?

A compound unit combines two measures with a division (or occasionally a multiplication), so converting it means converting the top and bottom separately, then recombining. For a density given in $\text{g/cm}^3$ that you need in $\text{kg/m}^3$ : convert grams to kilograms ( $\div 1000$ ) and cm³ to m³ ( $\div 1\,000\,000$ ) — since you're dividing the top by $1000$ and dividing the bottom by $1\,000\,000$ , the overall effect is multiplying the original number by $1000$ . Working the top and bottom out separately avoids trying to guess the combined factor.

Exam tip

Whenever a question mixes units (e.g. a length given in metres, but the answer wanted in cm²), convert all lengths to the same unit before calculating area or volume — converting the final area/volume answer with the wrong (unsquared/uncubed) factor is one of the most common lost marks at IGCSE.

06 — Worked example

Worked example

A rectangular tank measures $2\text{ m}$ by $150\text{ cm}$ by $80\text{ cm}$ . Find its volume in litres.

Convert every length to the same unit, cm: $2\text{ m} = 200\text{ cm}$ .
Volume $= 200 \times 150 \times 80 = 2\,400\,000 \text{ cm}^3$ .
Since $1\text{ cm}^3 = 1\text{ ml}$ : volume $= 2\,400\,000\text{ ml}$ .
Convert ml to litres ( $\div 1000$ ): volume $= 2400$ litres.

07 — Worked example

Worked example: converting a compound unit

A metal has a density of $2.7\text{ g/cm}^3$ . Express this density in $\text{kg/m}^3$ .

Convert the top (grams to kilograms): $2.7\text{ g} = 0.0027\text{ kg}$ .
Convert the bottom (cm³ to m³): $1\text{ cm}^3 = \dfrac{1}{1\,000\,000}\text{ m}^3$ , so dividing by this is the same as multiplying by $1\,000\,000$ .
Density $= 0.0027 \times 1\,000\,000 = 2700\text{ kg/m}^3$ .