Measurement Conversions

A recipe asks for cups, the hardware store sells lumber in feet, and the highway signs show kilometres. To make sense of all of them, you need to change from one unit to another. This page shows you how to convert within the metric system, how to handle square and cubic units (where most mistakes happen), and how to switch between metric and imperial units using a simple, reliable method called unit analysis.

Key ideas

Metric prefixes

Every metric unit is built from a base unit (metre for length, gram for mass, litre for capacity) and a prefix that tells you how many base units it’s worth.

Prefix	Symbol	Meaning	Examples
kilo	k	$1000$	$1 \text{ km} = 1000 \text{ m}$, $1 \text{ kg} = 1000 \text{ g}$
hecto	h	$100$	$1 \text{ hm} = 100 \text{ m}$
deca	da	$10$	$1 \text{ dam} = 10 \text{ m}$
(base unit)		$1$	metre, gram, litre
deci	d	$\dfrac{1}{10}$	$10 \text{ dm} = 1 \text{ m}$
centi	c	$\dfrac{1}{100}$	$100 \text{ cm} = 1 \text{ m}$
milli	m	$\dfrac{1}{1000}$	$1000 \text{ mm} = 1 \text{ m}$, $1000 \text{ mL} = 1 \text{ L}$

The ones you’ll use most are kilo, centi and milli. Each step on the ladder below is a factor of $10$.

From km to m is three steps down, so multiply by $10 \times 10 \times 10 = 1000$.

Which way: multiply or divide?

• Going to a smaller unit, you need more of them, so multiply. ($2 \text{ m} = 200 \text{ cm}$.)
• Going to a bigger unit, you need fewer of them, so divide. ($350 \text{ cm} = 3.5 \text{ m}$.)

Always ask: “Should my number get bigger or smaller?” That one question catches most mistakes.

Area units: square the factor

Area is measured in square units. A square that is $1$ cm on each side is also $10$ mm on each side, so its area is $10 \times 10 = 100 \text{ mm}^2$.

One square centimetre holds $100$ square millimetres, not $10$.

The same idea works for every area conversion. The length factor gets squared:

$$\begin{aligned} 1 \text{ cm}^2 &= 10 \times 10 = 100 \text{ mm}^2 \\ 1 \text{ m}^2 &= 100 \times 100 = 10\,000 \text{ cm}^2 \\ 1 \text{ km}^2 &= 1000 \times 1000 = 1\,000\,000 \text{ m}^2 \end{aligned}$$

Farm and park land in Canada is often measured in hectares: $1 \text{ ha} = 10\,000 \text{ m}^2$ (a square $100$ m on each side).

Volume and capacity units: cube the factor

Volume is measured in cubic units, so the length factor gets cubed:

$$\begin{aligned} 1 \text{ cm}^3 &= 10 \times 10 \times 10 = 1000 \text{ mm}^3 \\ 1 \text{ m}^3 &= 100 \times 100 \times 100 = 1\,000\,000 \text{ cm}^3 \end{aligned}$$

Capacity (how much a container holds) uses litres and millilitres. The link between volume and capacity is:

$$1 \text{ cm}^3 = 1 \text{ mL} \qquad 1000 \text{ cm}^3 = 1 \text{ L} \qquad 1 \text{ m}^3 = 1000 \text{ L}$$

Imperial units

Canada uses metric officially, but imperial units are still common in building, cooking and when you travel to the United States.

Imperial	Relationship
foot (ft)	$1 \text{ ft} = 12 \text{ in}$ (inches)
yard (yd)	$1 \text{ yd} = 3 \text{ ft}$
mile (mi)	$1 \text{ mi} = 5280 \text{ ft}$
pound (lb)	$1 \text{ lb} = 16 \text{ oz}$ (ounces)
US cup	$1 \text{ cup} = 8$ US fluid ounces

To switch systems, use a conversion factor. Questions will usually give you the one you need:

Conversion	Value
inches to centimetres	$1 \text{ in} = 2.54 \text{ cm}$ (exact)
feet to metres	$1 \text{ ft} = 30.48 \text{ cm} = 0.3048 \text{ m}$ (exact)
miles to kilometres	$1 \text{ mi} \approx 1.609 \text{ km}$
kilograms to pounds	$1 \text{ kg} \approx 2.205 \text{ lb}$
cups to millilitres	$1 \text{ cup} \approx 250 \text{ mL}$ (Canadian recipes)

(A US cup is a little smaller, about $237$ mL. For cooking, the difference rarely matters.)

Unit analysis

Unit analysis means multiplying by a fraction that equals $1$, with the units you want to get rid of on the bottom. The units cancel just like numbers do. For example, to change $5$ ft to inches:

$$5 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} = 60 \text{ in}$$

The fraction $\dfrac{12 \text{ in}}{1 \text{ ft}}$ equals $1$ because $12$ in and $1$ ft are the same length. Since ft is on the bottom, it cancels the ft you started with. If the units don’t cancel, flip the fraction. You can chain several fractions in a row for a multi-step conversion.

Where this comes from. Early units came from the human body. The ancient Egyptian cubit was the length of a forearm, from elbow to fingertip, and Egyptian builders used a standard royal cubit of about $52$ cm. The foot and the inch have similar origins. Because bodies vary, these units changed from place to place. The metric system was created in France in the 1790s so that everyone could share one set of units based on tens, and Canada began switching to metric in the 1970s. That’s why your grandparents might still give their height in feet and inches.

Worked examples

Example 1: Metric conversions

Convert.

• (a) $3.2$ km to metres
• (b) $450$ mm to centimetres
• (c) $2750$ g to kilograms
• (d) $0.6$ L to millilitres

Solution.

(a) Metres are smaller than kilometres, so multiply: $3.2 \times 1000 = 3200$ m.

(b) Centimetres are bigger than millimetres ($1 \text{ cm} = 10 \text{ mm}$), so divide: $450 \div 10 = 45$ cm.

(c) Kilograms are bigger, so divide: $2750 \div 1000 = 2.75$ kg.

(d) Millilitres are smaller, so multiply: $0.6 \times 1000 = 600$ mL.

Example 2: Converting an area

A bedroom floor is $4.5$ m long and $3.2$ m wide. Find its area in square metres and in square centimetres.

Solution. In square metres:

$$A = 4.5 \times 3.2 = 14.4 \text{ m}^2$$

Since $1 \text{ m}^2 = 10\,000 \text{ cm}^2$:

$$14.4 \text{ m}^2 \times \frac{10\,000 \text{ cm}^2}{1 \text{ m}^2} = 144\,000 \text{ cm}^2$$

Check another way: convert the lengths first. $4.5 \text{ m} = 450 \text{ cm}$ and $3.2 \text{ m} = 320 \text{ cm}$, so $A = 450 \times 320 = 144\,000 \text{ cm}^2$. ✓

Example 3: Imperial to metric

• (a) A TV screen is advertised as $55$ inches (measured along the diagonal). How many centimetres is that?
• (b) A basketball player is $6$ ft $1$ in tall. Find the height in centimetres, to the nearest centimetre.

Use $1 \text{ in} = 2.54 \text{ cm}$.

Solution.

(a)

$$55 \text{ in} \times \frac{2.54 \text{ cm}}{1 \text{ in}} = 139.7 \text{ cm}$$

(b) First write the whole height in inches: $6 \text{ ft} = 6 \times 12 = 72$ in, so the height is $72 + 1 = 73$ in. Then

$$73 \text{ in} \times \frac{2.54 \text{ cm}}{1 \text{ in}} = 185.42 \text{ cm} \approx 185 \text{ cm}$$

Example 4: Speed and capacity

• (a) On a road trip in the United States, the speed limit is $65$ mi/h. What is that in km/h, to the nearest whole number? Use $1 \text{ mi} \approx 1.609 \text{ km}$.
• (b) An aquarium is a rectangular box $60$ cm long, $30$ cm wide and $35$ cm tall. How many litres of water does it hold when full?

Solution.

(a) Only the distance unit changes, so one fraction does it:

$$65 \, \frac{\text{mi}}{\text{h}} \times \frac{1.609 \text{ km}}{1 \text{ mi}} = 104.585 \, \frac{\text{km}}{\text{h}} \approx 105 \text{ km/h}$$

That’s a little faster than the $100$ km/h limit on most stretches of Ontario’s 400-series highways.

(b) Volume of the box:

$$V = 60 \times 30 \times 35 = 63\,000 \text{ cm}^3$$

Since $1 \text{ cm}^3 = 1 \text{ mL}$, that’s $63\,000$ mL. Divide by $1000$ to get litres: $63$ L.

Common mistakes

Multiplying when you should divide (or the other way round). Before you calculate, decide whether the answer should be bigger or smaller. Changing $350$ cm to metres should give a smaller number ($3.5$ m), so if you got $35\,000$, you went the wrong way.

Using the length factor for area or volume. $1 \text{ m} = 100 \text{ cm}$, but $1 \text{ m}^2 = 10\,000 \text{ cm}^2$ and $1 \text{ m}^3 = 1\,000\,000 \text{ cm}^3$. Square the factor for area and cube it for volume. If you’re unsure, convert the lengths first and then calculate, as in the check in Example 2.

Thinking 1 cubic metre is 1 litre. A litre is only $1000 \text{ cm}^3$, a cube $10$ cm on each side. A cubic metre is much bigger: it holds $1000$ L.

Treating feet and inches like a decimal. $6$ ft $1$ in is not $6.1$ ft, because there are $12$ inches in a foot, not $10$. Change everything to one unit first ($73$ in).

Mixing units in one calculation. If a box is $1.2$ m long and $40$ cm wide, convert to the same unit before you multiply. $1.2 \times 40 = 48$ is meaningless; $120 \times 40 = 4800 \text{ cm}^2$ is right.

Rounding too early. Keep all the digits until the last step, then round once. Rounding $1.609$ to $2$ in the middle of a speed conversion would make your answer about $25\%$ too big.

Practice

1. (Warm-up) Convert.

• (a) $5.4$ m to centimetres
• (b) $82$ mm to centimetres
• (c) $3500$ m to kilometres

Solution

(a) Centimetres are smaller, so multiply by $100$: $540$ cm.

(b) Centimetres are bigger, so divide by $10$: $8.2$ cm.

(c) Kilometres are bigger, so divide by $1000$: $3.5$ km.

2. (Warm-up) Convert.

• (a) $1.25$ kg to grams
• (b) $375$ mL to litres

Solution

(a) $1.25 \times 1000 = 1250$ g.

(b) $375 \div 1000 = 0.375$ L.

3. (Core) A bread recipe needs $2.5$ cups of flour and $\dfrac{3}{4}$ cup of milk. Using $1 \text{ cup} = 250 \text{ mL}$, how many millilitres of each do you need?

Solution

Flour: $2.5 \times 250 = 625$ mL.

Milk: $\dfrac{3}{4} \times 250 = 187.5$ mL.

4. (Core)

• (a) A poster has an area of $0.8 \text{ m}^2$. What is its area in square centimetres?
• (b) A garden plot has an area of $25\,000 \text{ cm}^2$. What is its area in square metres?

Solution

Use $1 \text{ m}^2 = 10\,000 \text{ cm}^2$.

(a) Square centimetres are smaller, so multiply: $0.8 \times 10\,000 = 8000 \text{ cm}^2$.

(b) Square metres are bigger, so divide: $25\,000 \div 10\,000 = 2.5 \text{ m}^2$.

5. (Core) A piece of lumber at a building supply store is $8$ ft long. How long is it in centimetres? In metres, to two decimal places? Use $1 \text{ in} = 2.54 \text{ cm}$.

Solution

First change feet to inches, then inches to centimetres:

$$8 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} \times \frac{2.54 \text{ cm}}{1 \text{ in}} = 243.84 \text{ cm}$$

In metres: $243.84 \div 100 = 2.4384 \approx 2.44$ m.

6. (Core) The driving distance from Toronto to Montréal is about $540$ km. A visitor from the United States wants to know the distance in miles. Use $1 \text{ mi} \approx 1.609 \text{ km}$ and round to the nearest mile.

Solution

Put km on the bottom so it cancels:

$$540 \text{ km} \times \frac{1 \text{ mi}}{1.609 \text{ km}} \approx 335.6 \text{ mi} \approx 336 \text{ mi}$$

Check: miles are longer than kilometres, so the number of miles should be smaller. ✓

7. (Core) A hot tub is shaped like a rectangular box $2$ m long, $2$ m wide and $0.9$ m deep. How many litres of water does it hold when full?

Solution

$$V = 2 \times 2 \times 0.9 = 3.6 \text{ m}^3$$

Since $1 \text{ m}^3 = 1000 \text{ L}$:

$$3.6 \times 1000 = 3600 \text{ L}$$

8. (Challenge) A cyclist rides at $20$ metres per second down a long hill. Use unit analysis to convert this speed to kilometres per hour.

Solution

Change seconds to hours ($1 \text{ h} = 3600 \text{ s}$) and metres to kilometres ($1 \text{ km} = 1000 \text{ m}$), placing each unit so that it cancels:

$$20 \, \frac{\text{m}}{\text{s}} \times \frac{3600 \text{ s}}{1 \text{ h}} \times \frac{1 \text{ km}}{1000 \text{ m}} = \frac{20 \times 3600}{1000} \, \frac{\text{km}}{\text{h}} = 72 \text{ km/h}$$

9. (Challenge) A can of paint says it covers $350$ square feet. You want to paint walls with a total area of $30 \text{ m}^2$. Is one can enough? Use $1 \text{ ft} = 0.3048 \text{ m}$.

Solution

First find how big one square foot is in square metres. Square the length factor:

$$1 \text{ ft}^2 = 0.3048 \times 0.3048 = 0.09290304 \text{ m}^2$$

Now convert the wall area to square feet. Put $\text{m}^2$ on the bottom so it cancels:

$$30 \text{ m}^2 \times \frac{1 \text{ ft}^2}{0.09290304 \text{ m}^2} \approx 322.9 \text{ ft}^2$$

The walls are about $323 \text{ ft}^2$, which is less than $350 \text{ ft}^2$, so one can is enough (with a little to spare for touch-ups).