Standard Units & Conversions

1. The Standard (SI) Units

Scientists and mathematicians around the world use the International System of Units (SI) to avoid confusion. You should always aim to convert your values into these units before calculating.

Quantity Standard Unit Symbol Important Note
Length / Distance Metre m Base unit for area ($m^2$) and volume ($m^3$).
Mass Kilogram kg How much "stuff" is in an object.
Time Second s Convert hours and minutes to seconds for Physics formulas!
Weight (Force) Newton N Weight is a force caused by gravity. Weight = mass $\times$ gravity.
Temperature Celsius / Kelvin $^\circ\text{C}$ / K Everyday maths uses Celsius. Physics often uses Kelvin.
Mass vs. Weight Trap!
In everyday life, we say "I weigh 60 kg". Mathematically, this is wrong!
Your Mass is 60 kg. Your Weight is the force of gravity pulling you down.
Formula: $W = m \times g$ (where $g \approx 9.8$ or $10 \text{ N/kg}$ on Earth).
So, your weight is actually $60 \times 10 = 600 \text{ N}$.

2. The Common Prefixes

Prefixes are put in front of standard units to make them bigger or smaller without writing lots of zeros.

Prefix Symbol Meaning Multiplier (Standard Form) Example
Kilo k Thousand (1,000) $\times 10^3$ 1 km = 1,000 m
Centi c Hundredth ($\frac{1}{100}$) $\times 10^{-2}$ 100 cm = 1 m
Milli m Thousandth ($\frac{1}{1000}$) $\times 10^{-3}$ 1,000 mm = 1 m
Micro $\mu$ Millionth ($\frac{1}{1,000,000}$) $\times 10^{-6}$ $1,000,000 \mu\text{m}$ = 1 m

3. The Golden Rule: Convert First!

Convert before you calculate. If a question gives you distance in km and time in minutes, but asks for speed in m/s, change the distance to metres and the time to seconds before you divide.
Example 1: Time Conversion
Convert 2 hours and 15 minutes into seconds.
Step 1: Hours to minutes $2 \text{ hours} = 2 \times 60 = 120 \text{ mins}$
Total minutes = $120 + 15 = 135 \text{ mins}$.
Step 2: Minutes to seconds $135 \times 60 = 8,100 \text{ seconds}$
Example 2: Mass Conversion
Convert 4.5 grams (g) into kilograms (kg).
Logic: We are going from small (g) to big (kg), so the number gets smaller. Divide by 1,000.
$4.5 \div 1000 = 0.0045 \text{ kg}$

4. Compound Measures

Compound measures are made by combining two or more standard units.

Example 3: Compound Measure Calculation
A block has a mass of 450 g and a volume of $0.05 \text{ m}^3$. Calculate its density in kg/m$^3$.
Step 1: Check units! Volume is in $m^3$ (Good!). Mass is in g (Bad! We need kg).
$450 \text{ g} \div 1000 = 0.45 \text{ kg}$.
Step 2: Calculate Density $$\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{0.45}{0.05} = 9 \text{ kg/m}^3$$

Practice Questions & Solutions

Try these questions on paper, making sure to convert your units first, then click to reveal the worked solutions!

Q1: Convert 3.2 kilometres into metres.
"Kilo" means 1,000.
To go from km to m, multiply by 1,000.
$3.2 \times 1000 = 3200$
Answer: 3,200 m
Q2: Convert 450 milligrams (mg) into grams (g).
"Milli" means thousandth. There are 1,000 mg in 1 g.
To go from mg to g, divide by 1,000.
$450 \div 1000 = 0.45$
Answer: 0.45 g
Q3: Convert 1 hour and 40 minutes into seconds.
1 hour = 60 minutes.
Total minutes = $60 + 40 = 100 \text{ minutes}$.
There are 60 seconds in a minute.
$100 \times 60 = 6000$
Answer: 6,000 s
Q4: A car travels 12 km in 15 minutes. Calculate its speed in km/h.
Distance is already in km (12 km).
Time must be in hours. 15 minutes is $\frac{15}{60}$ hours, which is $0.25$ hours (or $\frac{1}{4}$ of an hour).
$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
$\text{Speed} = \frac{12}{0.25} = 48$
Answer: 48 km/h
Q5: A gold bar has a volume of $50 \text{ cm}^3$ and a density of $19.3 \text{ g/cm}^3$. Calculate its mass.
Formula: $\text{Mass} = \text{Density} \times \text{Volume}$
The units match up perfectly (cm$^3$ and g/cm$^3$), so no conversion is needed yet.
$\text{Mass} = 19.3 \times 50 = 965 \text{ g}$
Answer: 965 g (or 0.965 kg)
Q6: A biological cell has a diameter of 25 micrometres ($\mu\text{m}$). What is this in metres?
"Micro" means one millionth ($10^{-6}$).
To convert $\mu\text{m}$ to m, divide by 1,000,000.
$25 \div 1,000,000 = 0.000025$
In standard form: $2.5 \times 10^{-5}$
Answer: 0.000025 m (or $2.5 \times 10^{-5}$ m)
Q7: An astronaut has a mass of 75 kg. Calculate his weight on Earth (where $g = 9.8 \text{ N/kg}$).
Formula: $\text{Weight} = \text{mass} \times \text{gravity}$
$W = 75 \times 9.8$
$W = 735$
Answer: 735 N
Q8: TRICK QUESTION! Convert $2 \text{ m}^2$ into $\text{cm}^2$.
Warning: You do NOT just multiply by 100 for areas!
Think of a square that is $1\text{m} \times 1\text{m}$. Its area is $1 \text{ m}^2$.
In cm, that same square is $100\text{cm} \times 100\text{cm}$.
$100 \times 100 = 10,000 \text{ cm}^2$.
Therefore, $1 \text{ m}^2 = 10,000 \text{ cm}^2$.
So, $2 \text{ m}^2 = 2 \times 10,000 = 20,000$
Answer: 20,000 cm$^2$
Q9: Calculate the pressure exerted by a force of 600 N acting on an area of $5000 \text{ cm}^2$. Give your answer in N/m$^2$.
We want the answer in $\text{N/m}^2$, so we must convert the area from $\text{cm}^2$ to $\text{m}^2$ first.
From Q8, we know $1 \text{ m}^2 = 10,000 \text{ cm}^2$.
Area = $5000 \div 10,000 = 0.5 \text{ m}^2$.
$\text{Pressure} = \frac{\text{Force}}{\text{Area}}$
$\text{Pressure} = \frac{600}{0.5} = 1200$
Answer: 1,200 N/m$^2$
Q10: A cheetah runs 450 metres in 15 seconds. Calculate its speed in km/h.
Step 1: Convert units first.
Distance: $450 \text{ m} \div 1000 = 0.45 \text{ km}$.
Time: We need 15 seconds in hours.
Seconds to minutes: $15 \div 60 = 0.25 \text{ mins}$.
Minutes to hours: $0.25 \div 60 \approx 0.004166... \text{ hours}$ (It's easier to keep it as a fraction: $\frac{15}{3600}$ hours).

Step 2: Calculate Speed
$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
$\text{Speed} = 0.45 \div (\frac{15}{3600})$
$\text{Speed} = 0.45 \times \frac{3600}{15}$
$\text{Speed} = 108$

Alternative method: Find speed in m/s first ($450 \div 15 = 30 \text{ m/s}$). Then multiply by 3.6 to get km/h ($30 \times 3.6 = 108$).
Answer: 108 km/h