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Explore: Units, conversions and perimeter

Measurement · step 2 of 5

A question to start

In the 1790s, scientists in France wanted a unit of length that belonged to no king and no country. They chose the size of the Earth itself. They defined the metre as one ten-millionth of the distance from the North Pole to the equator.

That has a surprising result you can work out yourself. If the Pole to the equator is ten million metres, how far is it all the way round the Earth? (Answer: four times as far, 40 million metres, which is 40 000 kilometres. The real figure, measured today, is very close: about 40 075 km around the equator.)

The idea

The metric system

Most of the world measures with the metric system. Its big advantage is that every unit is linked to the next by 10, 100 or 1000, the same numbers our place-value system is built on. Each unit has a base name and a prefix that says how big it is:

Prefix Means Example
kilo- 1000 times 1 kilometre (km) = 1000 metres
(none) the base unit 1 metre (m)
centi- one hundredth 1 centimetre (cm) = \(\tfrac{1}{100}\) of a metre
milli- one thousandth 1 millimetre (mm) = \(\tfrac{1}{1000}\) of a metre

The same prefixes work for every kind of measure:

  • Length (how long): mm, cm, m, km. 10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km.
  • Mass (how much stuff, often called weight): grams (g) and kilograms (kg). 1000 g = 1 kg. A paper clip is about 1 g; a litre bottle of water is about 1 kg. 1000 kg = 1 tonne (t).
  • Capacity (how much a container holds): millilitres (mL) and litres (L). 1000 mL = 1 L. A teaspoon holds about 5 mL.

Converting between units

To change from one unit to another, ask: is the new unit bigger or smaller?

  • Going to a smaller unit, you need more of them, so multiply. 3.5 m in centimetres: each metre is 100 cm, so \(3.5 \times 100 = 350\) cm.
  • Going to a bigger unit, you need fewer of them, so divide. 4500 g in kilograms: every 1000 g is 1 kg, so \(4500 \div 1000 = 4.5\) kg.

Multiplying or dividing by 10, 100 or 1000 just moves the digits past the decimal point by 1, 2 or 3 places. That is why the metric system is so easy to convert.

Time is different

Time is the one measure that does not go in tens:

  • 60 seconds = 1 minute
  • 60 minutes = 1 hour
  • 24 hours = 1 day
  • 7 days = 1 week, and 365 days (366 in a leap year) = 1 year

The 60s go back to the ancient Babylonians, who counted in sixties. So be careful: 1.5 hours is not 1 hour 50 minutes. It is 1 hour and half an hour, which is 1 hour 30 minutes.

A day on Mars, a sol, lasts 24 hours 39 minutes 35 seconds. Mission teams who drive rovers sometimes live on "Mars time", starting work about 40 minutes later each Earth day to keep up with the rover's sunrise.

Perimeter

The perimeter of a flat shape is the total distance around its edge. Imagine an ant walking once around the outside: the perimeter is how far it walks. To find it, add up the lengths of all the sides.

For a rectangle with length \(l\) and width \(w\), there are two of each:

\[\text{perimeter of a rectangle} = 2 \times (l + w)\]

A square with side \(s\) has four equal sides, so its perimeter is \(4 \times s\).

Perimeter is a length, so it is measured in length units: cm, m, km.

Choosing sensible units

Use a unit that gives a number that is easy to picture. The length of an ant: millimetres. A pencil: centimetres. A room: metres. A road trip: kilometres. Saying "the trip is 120 000 000 mm" is correct but not useful.

Worked example

A garden is shaped like a capital L. Find its perimeter.

10 m 5 m 4 m 3 m 6 m 8 m
  1. Go round the shape and list every side: bottom 10 m, right 5 m, step 4 m, up 3 m, top 6 m, left 8 m.
  2. Check the missing lengths make sense: the left side (8 m) should equal the right side plus the step up, \(5 + 3 = 8\) ✓. The bottom (10 m) should equal the top plus the step across, \(6 + 4 = 10\) ✓.
  3. Add them all: \(10 + 5 + 4 + 3 + 6 + 8 = 36\) m.

Here is something curious: a plain 10 m by 8 m rectangle also has perimeter \(2 \times (10 + 8) = 36\) m. Cutting a corner out of a rectangle inwards like this removes two sides but adds two equal ones, so the perimeter stays the same, even though the garden is smaller.

And a time one: how many minutes longer is a sol than an Earth day?

  1. A sol is 24 h 39 min 35 s; an Earth day is 24 h.
  2. The difference is 39 min 35 s, almost 40 minutes.
  3. Over 30 sols that adds up to \(30 \times 39.6 \approx 1188\) minutes, nearly 20 hours. A month of living on Mars time puts you most of a day out of step with Earth.

Try it

  1. Write 3.5 m in centimetres, and 2.4 km in metres.
  2. Write 4500 g in kilograms, and 750 mL in litres.
  3. Find the perimeter of a rectangle 12 cm long and 7.5 cm wide.
  4. A square has a perimeter of 36 cm. How long is each side?
  5. A film lasts 150 minutes. Write that in hours and minutes.
  6. A rectangle has perimeter 30 m and length 9 m. How wide is it?
Answers
  1. \(3.5 \times 100 = 350\) cm. \(2.4 \times 1000 = 2400\) m.
  2. \(4500 \div 1000 = 4.5\) kg. \(750 \div 1000 = 0.75\) L.
  3. \(2 \times (12 + 7.5) = 2 \times 19.5 = 39\) cm.
  4. Four equal sides share 36 cm, so each is \(36 \div 4 = 9\) cm.
  5. 120 minutes is 2 hours, with 30 minutes left over: 2 hours 30 minutes (or 2.5 hours).
  6. Half the perimeter is one length plus one width: \(30 \div 2 = 15\) m. Take away the length: \(15 - 9 = 6\) m.

Common mistake

Mixing units in one sum. A rectangle 2 m long and 50 cm wide does not have perimeter \(2 \times (2 + 50) = 104\). Change everything to the same unit first: 50 cm is 0.5 m, so the perimeter is \(2 \times (2 + 0.5) = 5\) m (or, in centimetres, \(2 \times (200 + 50) = 500\) cm). Always write the unit next to each number; a sum with mixed units then looks wrong straight away.

Where it leads