Measures, units and scale drawings

Q: What is the general rule for converting between larger and smaller metric units?

To convert from a bigger unit to a smaller unit , multiply by the conversion factor; to convert from a smaller unit to a bigger unit , divide .

Mathematics 2210 · CCEA GCSE · 7 min read · Updated: 31 Aug 2026

Welcome to Measures, Units, and Scale Drawings!

Have you ever tried baking a cake, planning a road trip, assembling flat-pack furniture, or reading a video game map? If so, you have already used measures and scales! In this chapter, we will master how to measure, convert, and calculate real-life quantities like length, speed, mass, and scale representations. Don't worry if converting units has felt confusing in the past — by breaking it down step-by-step and using simple visual tricks, you will be solving these questions with confidence!

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1. Metric Units and Standard Conversions

The metric system is based on powers of $10$, which makes it very neat once you remember the prefixes:

• Milli- means one-thousandth ($\frac{1}{1000}$)
• Centi- means one-hundredth ($\frac{1}{100}$)
• Kilo- means one thousand ($1000$)

A. Length

The standard metric units of length are millimetres ($\text{mm}$), centimetres ($\text{cm}$), metres ($\text{m}$), and kilometres ($\text{km}$).

• $1\text{ cm} = 10\text{ mm}$
• $1\text{ m} = 100\text{ cm}$
• $1\text{ km} = 1000\text{ m}$

B. Mass (Weight)

Mass tells us how heavy an object is. The units are milligrams ($\text{mg}$), grams ($\text{g}$), kilograms ($\text{kg}$), and tonnes ($\text{t}$).

• $1\text{ g} = 1000\text{ mg}$
• $1\text{ kg} = 1000\text{ g}$
• $1\text{ tonne} = 1000\text{ kg}$

C. Capacity and Liquid Volume

Capacity measures how much liquid a container holds. The units are millilitres ($\text{ml}$), centilitres ($\text{cl}$), and litres ($\text{l}$).

• $1\text{ cl} = 10\text{ ml}$
• $1\text{ l} = 1000\text{ ml} = 100\text{ cl}$
• Useful link to 3D shapes: $1\text{ ml} = 1\text{ cm}^3$ and $1\text{ l} = 1000\text{ cm}^3$

The "Big-to-Small" Conversion Rule

A quick rule to remember whether to multiply or divide:

• Going from a BIG unit to a SMALL unit $\rightarrow$ MULTIPLY (e.g., $\text{m} \rightarrow \text{cm}$: $3.5\text{ m} \times 100 = 350\text{ cm}$)
• Going from a SMALL unit to a BIG unit $\rightarrow$ DIVIDE (e.g., $\text{g} \rightarrow \text{kg}$: $4500\text{ g} \div 1000 = 4.5\text{ kg}$)

Key Takeaway: When you convert to a smaller unit, you need a bigger number of them to make the same amount, so you multiply.

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2. Area and Volume Unit Conversions (Watch Out!)

A very common mistake in GCSE exams is thinking that $1\text{ m}^2 = 100\text{ cm}^2$. Let's see why that is not true!

Converting Area Units ($\text{cm}^2 \leftrightarrow \text{m}^2$)

Imagine a square floor measuring $1\text{ m}$ by $1\text{ m}$:
• The area in metres is $1\text{ m} \times 1\text{ m} = 1\text{ m}^2$.
• In centimetres, each side is $100\text{ cm}$.
• So the area in centimetres is $100\text{ cm} \times 100\text{ cm} = 10\,000\text{ cm}^2$!

Rule for Area: Apply the length conversion factor twice (square it):
• To convert $\text{m}^2$ to $\text{cm}^2$: Multiply by $100^2 = 10\,000$.
• To convert $\text{cm}^2$ to $\text{mm}^2$: Multiply by $10^2 = 100$.

Converting Volume Units ($\text{cm}^3 \leftrightarrow \text{m}^3$)

Imagine a cube measuring $1\text{ m} \times 1\text{ m} \times 1\text{ m}$:
• The volume is $1\text{ m}^3$.
• In centimetres, this is $100\text{ cm} \times 100\text{ cm} \times 100\text{ cm} = 1\,000\,000\text{ cm}^3$!

Rule for Volume: Apply the length conversion factor three times (cube it):
• To convert $\text{m}^3$ to $\text{cm}^3$: Multiply by $100^3 = 1\,000\,000$.
• To convert $\text{cm}^3$ to $\text{mm}^3$: Multiply by $10^3 = 1000$.

Common Mistake to Avoid: Never just multiply by $100$ when dealing with area or volume. Always square the conversion for area, and cube it for volume!

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3. Imperial Units and Approximations

In Northern Ireland and the UK, imperial units are still used in daily life (like road signs in miles and milk in pints). You should know these rough metric equivalents:

• Miles and Kilometres: $5\text{ miles} \approx 8\text{ km}$ (or $1\text{ mile} \approx 1.6\text{ km}$)
• Inches and Centimetres: $1\text{ inch} \approx 2.5\text{ cm}$
• Feet and Centimetres: $1\text{ foot} \approx 30\text{ cm}$
• Kilograms and Pounds: $1\text{ kg} \approx 2.2\text{ pounds (lbs)}$
• Litres and Pints: $1\text{ litre} \approx 1.75\text{ pints}$ (or $1\text{ pint} \approx 570\text{ ml}$)
• Gallons and Litres: $1\text{ gallon} \approx 4.5\text{ litres}$

Worked Example: Miles to Kilometres

Question: A road trip is $65\text{ miles}$. How far is this in kilometres?

Step 1: Remember that $5\text{ miles} \approx 8\text{ km}$.
Step 2: Find what $1\text{ mile}$ is worth: $8 \div 5 = 1.6\text{ km}$.
Step 3: Multiply by $65$: $65 \times 1.6 = 104\text{ km}$.
Answer: $104\text{ km}$

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4. Compound Measures

A compound measure is created by combining two or more different measures. The three most important ones are Speed, Density, and Pressure.

A. Speed, Distance, and Time

Speed is the distance travelled per unit of time.
$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

• $\text{Distance} = \text{Speed} \times \text{Time}$
• $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Crucial Tip on Converting Time to Decimals:
Time is based on $60$ minutes, not $100$! To convert minutes to a decimal hour, divide by $60$:
• $15\text{ minutes} = \frac{15}{60} = 0.25\text{ hours}$
• $30\text{ minutes} = \frac{30}{60} = 0.5\text{ hours}$
• $45\text{ minutes} = \frac{45}{60} = 0.75\text{ hours}$
• $2\text{ hours } 18\text{ minutes} = 2 + \frac{18}{60} = 2.3\text{ hours}$ (not $2.18$ hours!)

B. Density, Mass, and Volume

Density tells you how tightly packed the matter inside an object is (mass per unit volume).
$$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$$

• $\text{Mass} = \text{Density} \times \text{Volume}$
• $\text{Volume} = \frac{\text{Mass}}{\text{Density}}$
• Common units: $\text{g/cm}^3$ or $\text{kg/m}^3$.

C. Pressure, Force, and Area

Pressure measures how much force is applied over a given area of surface.
$$\text{Pressure} = \frac{\text{Force}}{\text{Area}}$$

• $\text{Force} = \text{Pressure} \times \text{Area}$
• $\text{Area} = \frac{\text{Force}}{\text{Pressure}}$
• Common units: $\text{N/m}^2$ (also called Pascals, $\text{Pa}$) or $\text{N/cm}^2$.

Key Takeaway: For all compound measures, the units in the question tell you the formula! For example, speed in $\text{km/h}$ means $\text{kilometres (distance)} \div \text{hours (time)}$.

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5. Scale Drawings and Maps

Scale drawings allow us to represent large real-world objects (like buildings or towns) accurately on paper.

Understanding Scales as Ratios

A map scale is usually given as a ratio like $1 : 50\,000$.
This means: $1\text{ unit}$ on the map represents $50\,000$ of the same units in real life.

Step-by-Step Method for Map Scales

Example: A map has a scale of $1 : 25\,000$. Two points on the map are $6\text{ cm}$ apart. What is the real-life distance in kilometres?

Step 1: Multiply map distance by the scale factor:
$$\text{Real distance} = 6\text{ cm} \times 25\,000 = 150\,000\text{ cm}$$

Step 2: Convert centimetres to metres (divide by $100$):
$$150\,000\text{ cm} \div 100 = 1500\text{ m}$$

Step 3: Convert metres to kilometres (divide by $1000$):
$$1500\text{ m} \div 1000 = 1.5\text{ km}$$

Answer: The actual distance is $1.5\text{ km}$.

Going in Reverse: Real Life to Map

Example: A park path is $400\text{ m}$ long in real life. How long will it be on a drawing with a scale of $1 : 2000$?

Step 1: Convert the real distance to centimetres first:
$$400\text{ m} \times 100 = 40\,000\text{ cm}$$

Step 2: Divide by the scale factor:
$$40\,000\text{ cm} \div 2000 = 20\text{ cm}$$

Answer: The line on the drawing should be $20\text{ cm}$ long.

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6. Reading Scales and Measuring Instruments

Whether you are reading a speedometer, a thermometer, or a measuring jug, the process is always the same:

Step 1: Find two numbered values next to each other on the scale and subtract them to find the total difference.
Step 2: Count how many small intervals (spaces) there are between those two numbers.
Step 3: Divide the difference by the number of intervals to find what each individual mark is worth.
Step 4: Count up from the nearest marked number to where the pointer or liquid level is.

Quick Example: The markings on a cylinder show $200\text{ ml}$ and $300\text{ ml}$. There are $5$ small intervals between them.
• Difference $= 300 - 200 = 100\text{ ml}$
• Value of each mark $= 100 \div 5 = 20\text{ ml}$
• If the level is $3$ marks above $200\text{ ml}$, the reading is $200 + (3 \times 20) = 260\text{ ml}$.

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Quick Review & Top Exam Tips

• Check your units first: Always convert all lengths, times, or masses into the same units before calculating.
• Time in decimal form: Always divide minutes by $60$ to get hours (e.g., $36\text{ mins} = \frac{36}{60} = 0.6\text{ hours}$).
• Units of area and volume: Remember to square the scale factor for area ($\times 100^2$ for $\text{m}^2 \rightarrow \text{cm}^2$) and cube it for volume ($\times 100^3$ for $\text{m}^3 \rightarrow \text{cm}^3$).
• Scale drawing rule of thumb: Real life $\rightarrow$ Map = DIVIDE. Map $\rightarrow$ Real life = MULTIPLY.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the general rule for converting between larger and smaller metric units?

To convert from a bigger unit to a smaller unit, multiply by the conversion factor; to convert from a smaller unit to a bigger unit, divide.

Why is $1\text{ m}^2$ equal to $10\,000\text{ cm}^2$ rather than $100\text{ cm}^2$?

Because area has two dimensions, the linear conversion factor must be applied twice (squared): $1\text{ m}^2 = 100\text{ cm} \times 100\text{ cm} = 10\,000\text{ cm}^2$.

How do you convert between cubic metres ($\text{m}^3$) and cubic centimetres ($\text{cm}^3$)?

Cube the linear conversion factor: multiply by $100^3 = 1\,000\,000$ to go from $\text{m}^3$ to $\text{cm}^3$, or divide by $1\,000\,000$ to go from $\text{cm}^3$ to $\text{m}^3$.

What is the key approximation used to convert between miles and kilometres?

$5\text{ miles} \approx 8\text{ km}$ (or $1\text{ mile} \approx 1.6\text{ km}$).

How do you convert minutes into decimal hours for speed, distance, and time calculations?

Divide the number of minutes by $60$ (e.g., $18\text{ minutes} = \frac{18}{60} = 0.3\text{ hours}$).

What is the formula relating density, mass, and volume?

$$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$$

What is the formula relating pressure, force, and area?

$$\text{Pressure} = \frac{\text{Force}}{\text{Area}}$$

What does a map scale ratio such as $1 : 50\,000$ mean?

It means that $1\text{ unit}$ measured on the map represents $50\,000$ of the exact same units in real life.

How do you find the value of each interval mark on an analogue measuring scale?

Subtract two adjacent numbered values to get the total difference, then divide by the number of small subdivisions (intervals) between them.

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