Units of Measurement (HSC SSCE Mathematics Standard): Revision Notes

Units of Measurement

What is measurement?

When we need to find out how big something is or how much of something we have, we use measurement. This process involves determining the size or amount of a quantity. To measure accurately, we typically need a measuring instrument. Different tools are used for measuring different things. For example, when measuring length or distance, you might use a ruler, tape measure, or even GPS technology depending on what you're measuring and how precise you need to be.

There are several different measurement systems used around the world, but in Australia we use the SI metric system. This system makes calculations and conversions much easier because it's based on powers of ten.

The SI metric system

The letters 'SI' stand for Système International, which is French for International System of Units. This system is used by most countries worldwide and is the standard for scientific work. The key advantage of the SI system is that it's based on multiples of $10$, which makes converting between units straightforward. You can easily multiply or divide to change from one unit to another.

In the SI system, some units are official SI units (the standard units), while others are non-SI units that are approved for everyday use alongside the official units. Understanding which is which helps you know when you're working with the formal scientific standard.

SI units for different quantities

The SI system has base units for measuring different types of quantities. A base unit is the fundamental unit from which other units are derived. Let's look at the main quantities you'll work with:

This table shows you the base units and the most commonly used derived units for length, area, volume, and mass. Notice how each quantity has one base unit, and then several other units that relate to it through powers of ten.

Length units

The base unit for length is the metre (symbol: $\text{m}$). Other length units include:

• Millimetre ($\text{mm}$): $1000 \text{ mm} = 1 \text{ m}$
• Centimetre ($\text{cm}$): $100 \text{ cm} = 1 \text{ m}$
• Kilometre ($\text{km}$): $1 \text{ km} = 1000 \text{ m}$

Area units

The base unit for area is the square metre (symbol: $\text{m}^2$). This measures two-dimensional space. Other area units include:

• Square centimetre ($\text{cm}^2$): $10000 \text{ cm}^2 = 1 \text{ m}^2$
• Hectare ($\text{ha}$): $1 \text{ ha} = 10000 \text{ m}^2$

Volume units

The base unit for volume is the cubic metre (symbol: $\text{m}^3$). This measures three-dimensional space. Other volume units include:

• Cubic centimetre ($\text{cm}^3$): $1000000 \text{ cm}^3 = 1 \text{ m}^3$
• Litre ($\text{L}$): $1 \text{ L} = 1000 \text{ cm}^3$
• Millilitre ($\text{mL}$): $1000 \text{ mL} = 1 \text{ L}$
• Kilolitre ($\text{kL}$): $1 \text{ kL} = 1000 \text{ L}$

Mass units

The base unit for mass is the kilogram (symbol: $\text{kg}$). Other mass units include:

• Gram ($\text{g}$): $1000 \text{ g} = 1 \text{ kg}$
• Milligram ($\text{mg}$): $1000 \text{ mg} = 1 \text{ g}$
• Tonne ($\text{t}$): $1 \text{ t} = 1000 \text{ kg}$

Understanding mass

It's important to understand what mass actually measures. Mass is the amount of matter (stuff) that makes up an object or body. This is different from weight, even though we often use these words interchangeably in everyday life.

Weight is actually the force that gravity exerts on a mass. Here's the key difference: if you travel to a different location where gravity is stronger or weaker (like the Moon or Jupiter), your mass stays exactly the same, but your weight changes. This is because you still have the same amount of matter in your body, but the gravitational force acting on that matter is different.

However, when you're on Earth's surface and not moving, mass and weight are considered equivalent for everyday purposes. This is why we can use scales to "weigh" objects and report the result in kilograms, which is technically a unit of mass.

Converting between SI units of the same type

One of the most useful features of the SI system is how easy it is to convert between units. The prefix attached to a unit tells you exactly how that unit relates to the base unit. This prefix indicates multiplication by a power of $10$.

The most common prefixes you'll encounter are:

• Mega: means multiply by $1000000$ (or $10^6$)
• Kilo: means multiply by $1000$ (or $10^3$)
• Centi: means divide by $100$ (or multiply by $\frac{1}{100}$ or $10^{-2}$)
• Milli: means divide by $1000$ (or multiply by $\frac{1}{1000}$ or $10^{-3}$)

The diagram below shows you how this works:

Converting length units

Let's see how this works with length conversions.

Converting mass units

The same principles apply when converting mass units.

Converting area units

Converting area units is slightly different from converting length units. This is because area is two-dimensional, so the conversion factor must be squared.

To understand this, let's look at converting square metres to square centimetres. We start by converting the side lengths:

This diagram shows that a square with sides of $1 \text{ m}$ is equal to a square with sides of $100 \text{ cm}$.

Now let's calculate the areas:

For the square in metres: $\text{Area} = 1 \text{ m} \times 1 \text{ m} = 1 \text{ m}^2$

For the square in centimetres: $\text{Area} = 100 \text{ cm} \times 100 \text{ cm} = 10000 \text{ cm}^2$

Therefore: $1 \text{ m}^2 = 10000 \text{ cm}^2$

Or equivalently: $1 \text{ cm}^2 = \frac{1}{10000} \text{ m}^2$

Notice that even though $1 \text{ m} = 100 \text{ cm}$ (a factor of $100$), we have $1 \text{ m}^2 = 10000 \text{ cm}^2$ (a factor of $100^2 = 10000$). This is because we're multiplying both the length and the width by $100$.

Converting volume units

Converting volume units requires even more care because volume is three-dimensional, so the conversion factor must be cubed.

Let's look at converting cubic metres to cubic centimetres:

This diagram shows that a cube with edges of $1 \text{ m}$ is equal to a cube with edges of $100 \text{ cm}$.

Now let's calculate the volumes:

For the cube in metres: $\text{Volume} = 1 \text{ m} \times 1 \text{ m} \times 1 \text{ m} = 1 \text{ m}^3$

For the cube in centimetres: $\text{Volume} = 100 \text{ cm} \times 100 \text{ cm} \times 100 \text{ cm} = 1000000 \text{ cm}^3$

Therefore: $1 \text{ m}^3 = 1000000 \text{ cm}^3$

Or equivalently: $1 \text{ cm}^3 = \frac{1}{1000000} \text{ m}^3$

Even though $1 \text{ m} = 100 \text{ cm}$ (a factor of $100$), we have $1 \text{ m}^3 = 1000000 \text{ cm}^3$ (a factor of $100^3 = 1000000$). This is because we're multiplying the length, width, and height each by $100$.