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Imperial units in Canada

Measurement · level 5, lesson 6 of 6

A question to start

In September 1999 a spacecraft called Mars Climate Orbiter reached Mars after a journey of more than nine months, fired its engine to slip into orbit, and was never heard from again. It had come in far too low and was lost in the Martian air.

The cause, found by NASA's investigation board, was not a broken part. One piece of software reported the push of the spacecraft's thrusters in pound-force seconds, an imperial unit, and another piece of software read the numbers as newton-seconds, the metric unit. One pound-force is about 4.45 newtons, so the thruster effects were counted as less than a quarter of what they really were, and small errors built up over months of flight.

Units are not decoration. In Canada we live with two systems every day, the metric system and the older imperial units, and this lesson shows how to move between them safely.

The idea

Two systems

The metric system (officially the SI, from the French Système international d'unités) is built on tens: 10 mm in a cm, 100 cm in a m, 1000 m in a km. You have used it in every lesson so far.

The imperial units are older. They grew from everyday measures (a foot, a thumb's width, a day's ploughing) and were standardised in Britain. The United States uses a close relative, called US customary units. The main ones:

Kind Unit In smaller units Metric value
length inch (in) 2.54 cm exactly
length foot (ft) 12 in 30.48 cm
length yard (yd) 3 ft = 36 in 91.44 cm
length mile (mi) 1760 yd = 5280 ft about 1.609 km
mass ounce (oz) about 28.35 g
mass pound (lb) 16 oz about 0.4536 kg
volume US cup about 237 mL
volume US gallon about 3.785 L
volume imperial gallon about 4.546 L

The numbers in the "in smaller units" column are not powers of ten: 12, 3, 1760, 16. That is the main reason the imperial system is harder to calculate with.

Why "2.54 exactly"?

Since 1959, Canada, the United States, Britain and several other countries have defined the inch to be exactly 2.54 cm, and the pound to be exactly 0.453 592 37 kg. Before that agreement, the American and British inches were very slightly different. So the inch is no longer a separate unit with its own standard: it is a name for 2.54 cm. Every other imperial length follows from it exactly: a foot is \(12 \times 2.54 = 30.48\) cm, and a mile is \(5280 \times 30.48 = 160\,934.4\) cm \(= 1.609\,344\) km.

0123 4567 891011 12 0123 4 inches centimetres

Why Canada uses both

Canada began changing over to the metric system in the 1970s. Road signs changed to kilometres and kilometres per hour in 1977, and today fuel is sold in litres, weather is given in degrees Celsius, and food is labelled in grams and millilitres. But imperial units never fully left:

  • many people give their height in feet and inches and their weight in pounds;
  • houses and apartments are often advertised in square feet, and lumber is sold as "2 by 4s" (whose real size, after drying and planing, is about 1.5 in by 3.5 in);
  • many recipes use cups and teaspoons, and many ovens are marked in degrees Fahrenheit;
  • Canada's biggest trading partner, the United States, uses customary units, so many products and machines come with them.

A sensible Canadian can think in both and convert carefully. Notice one more trap: a "cup" in a Canadian recipe today usually means a metric cup of 250 mL, while an American cup is about 237 mL. And an imperial gallon (used in Britain and in older Canadian measures) is about 20% bigger than a US gallon.

Converting with "a fraction equal to 1"

Here is a safe method that works for every conversion. Since 1 in = 2.54 cm, the fraction

\[\frac{2.54 \text{ cm}}{1 \text{ in}}\]

has the same length on top and on the bottom, so it is equal to 1. Multiplying by 1 does not change a length, it only changes its name. Choose the fraction so that the unit you want to get rid of cancels:

\[67 \text{ in} \times \frac{2.54 \text{ cm}}{1 \text{ in}} = 170.18 \text{ cm}\]

The "in" on top and the "in" on the bottom cancel, leaving cm. To go the other way, turn the fraction upside down:

\[100 \text{ cm} \times \frac{1 \text{ in}}{2.54 \text{ cm}} \approx 39.37 \text{ in}\]

You can chain several fractions: feet to inches to centimetres to metres. If the units do not cancel neatly, the fraction is upside down. This trick is called dimensional analysis, and you will use it all through science.

Areas and volumes: square the factor again

Everything from lesson 5 still holds. Since 1 ft = 0.3048 m,

\[1 \text{ ft}^2 = 0.3048^2 \text{ m}^2 \approx 0.0929 \text{ m}^2\]

so a square metre is about 10.8 square feet. One acre (a farm unit) is about 0.405 hectares.

Temperature is different

Celsius and Fahrenheit are not related by just multiplying: their zeros are in different places. Water freezes at 0 °C, which is 32 °F, and boils at 100 °C, which is 212 °F. So 100 Celsius degrees span 180 Fahrenheit degrees, and each Celsius degree is \(\tfrac{180}{100} = \tfrac95\) of a Fahrenheit degree:

\[F = \tfrac95 C + 32 \qquad C = \tfrac59 (F - 32)\]

Worked examples

1. A height. Someone is 5 ft 7 in tall. Are they taller than 1.7 m?

  1. All in inches: \(5 \times 12 + 7 = 67\) in.
  2. \(67 \times 2.54 = 170.18\) cm, which is 1.7018 m. Just taller than 1.7 m, by about 2 mm.

2. A marathon. A marathon is 42.195 km. How many miles?

\(42.195 \text{ km} \times \dfrac{1 \text{ mi}}{1.609\,344 \text{ km}} \approx 26.22\) miles. (Runners call it "26.2".)

3. A newborn baby. A baby weighs 7 lb 8 oz. What is that in kilograms?

  1. 8 oz is \(\tfrac{8}{16} = 0.5\) lb, so the baby weighs 7.5 lb.
  2. \(7.5 \times 0.453\,592\,37 \approx 3.40\) kg.

4. A house. An apartment is advertised as 1500 square feet. About how many square metres?

  1. 1 ft = 0.3048 m, so 1 ft² = \(0.3048^2 = 0.092\,903\) m².
  2. \(1500 \times 0.092\,903 \approx 139.4\) m².

5. An oven. A recipe from the United States says 350 °F. What is that in Celsius?

\(C = \tfrac59 (350 - 32) = \tfrac59 \times 318 \approx 176.7\) °C. Most Canadian recipes round this to 180 °C.

6. Speed. A US road sign says 60 mph (miles per hour). In km/h: \(60 \times 1.609\,344 \approx 96.6\) km/h.

Try it

  1. A ruler is 12 inches long. How long is it in centimetres?
  2. How many inches are there in 1 m, to two decimal places?
  3. A basketball player is 6 ft 2 in tall. What is that in centimetres?
  4. A highway sign says 100 km/h. What is that in miles per hour, to the nearest whole number?
  5. A bag of flour weighs 2 kg. About how many pounds is that?
  6. A "2 by 4" is really about 1.5 in by 3.5 in. Find the true area of its end in square inches and in cm².
  7. A bedroom is 12 ft by 15 ft. Find its area in square feet and in square metres.
  8. Mars's average surface temperature is about −60 °C. What is that in Fahrenheit? And at what temperature do the two scales show the same number?
Answers
  1. \(12 \times 2.54 = 30.48\) cm.
  2. \(100 \div 2.54 \approx 39.37\) in.
  3. \(6 \times 12 + 2 = 74\) in, and \(74 \times 2.54 = 187.96\) cm, about 188 cm.
  4. \(100 \div 1.609\,344 \approx 62.1\), so about 62 mph.
  5. \(2 \div 0.453\,592\,37 \approx 4.41\) lb.
  6. \(1.5 \times 3.5 = 5.25\) in². Each in² is \(2.54^2 = 6.4516\) cm², so \(5.25 \times 6.4516 \approx 33.9\) cm². (Or lengths first: \(3.81 \text{ cm} \times 8.89 \text{ cm} \approx 33.9\) cm².)
  7. \(12 \times 15 = 180\) ft². \(180 \times 0.092\,903 \approx 16.7\) m².
  8. \(F = \tfrac95 \times (-60) + 32 = -108 + 32 = -76\) °F. The scales agree when \(F = C\): \(C = \tfrac95 C + 32\), so \(-\tfrac45 C = 32\) and \(C = -40\). At −40 the two scales read the same.

Common mistake

Dropping the units. Writing "67 × 2.54 = 170.18" with no units hides whether you went inches to cm or cm to inches. Write the units in every line and make them cancel. That is exactly the habit that would have saved Mars Climate Orbiter.

Treating imperial units as if they were decimal. 5 ft 7 in is not 5.7 ft: it is \(5 + \tfrac{7}{12} \approx 5.58\) ft. In the same way, 7 lb 8 oz is 7.5 lb, but 7 lb 4 oz is 7.25 lb, not 7.4 lb.

Where it leads