Measurement — Print
Every conversion is a multiplication by one. Write the unit you want on top, the unit you have on the bottom, and the wrong units cancel themselves.
What the practice questions cover
These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Area — Space inside, measured in square units.
- Base — The number being multiplied. In 53 the base is 5.
- Factor — A number that divides evenly into another.
- Parallel — Same slope, different intercept.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
- Trapezoid — A quadrilateral with exactly one pair of parallel sides.
- Dimensions — Rows × columns. A 2 × 3 matrix has two rows and three columns.
- Height — The perpendicular distance from the base to the opposite corner.
Measure a table with a centimetre ruler and you get 150. Measure the same table in metres and you get 1.5.
The table did not shrink. Only the size of the unit changed.
A centimetre is small, so it takes a lot of them — a big number. A metre is big, so it takes few — a small number.
That gives you a check you can apply every time: converting to a smaller unit must give a bigger number.
If you convert metres to centimetres and your answer gets smaller, you multiplied when you should have divided.
Set conversions up as fractions and let the units cancel: 150 cm × 1 m100 cm. If the unit you want to lose does not cancel, the fraction is upside down — and you will know before you do any arithmetic.
Converting by cancelling units
A conversion factor is a fraction worth exactly 1, because top and bottom are the same amount: 12 in1 ft = 1.
Multiplying by 1 never changes the amount — only how it is written.
Convert 7 feet to inches: 7 ft × 12 in1 ft = 84 in. The "ft" cancels, leaving inches.
Which way up?
Put the unit you are getting rid of on the bottom, so it cancels. If your answer comes out with a strange unit like ft²/in, you had the fraction upside down.
The metric system is just powers of ten
km, m, cm, mm — each step is a factor of 10, 100 or 1000, so converting is moving the decimal point.
1 m = 100 cm, 1 km = 1000 m, 1 cm = 10 mm.
Going to a smaller unit gives a bigger number: 2.5 m is 250 cm. If your number shrank when the unit shrank, you went the wrong way.
Square and cubic units do not convert the same way
1 m = 100 cm, but 1 m2 is not 100 cm2.
A square metre is 100 cm by 100 cm, so 1 m2 = 10,000 cm2. The factor is squared.
Likewise 1 m3 = 1,000,000 cm3 — the factor is cubed. This is the same k, k2, k3 idea as scaling.
The classic trap
Converting an area by the plain length factor. 5 m2 is 50,000 cm2, not 500 cm2.
Perimeter and area, quickly
Rectangle: P = 2(l + w), A = lw.
Triangle: A = 12bh, using the perpendicular height.
Parallelogram: A = bh. Trapezoid: A = 12(b1 + b2)h.
For an odd shape, split it into pieces you recognise, find each area, and add — or find a big shape and subtract what is missing.
Choosing a sensible unit
A room is measured in metres, a pencil in centimetres, a city in kilometres. Choosing badly makes numbers you cannot check.
Before you write an answer, ask whether it is plausible. A desk with an area of 2 km2 should stop you.
Worked examples
Two-step conversion
Convert 3 miles to inches. (1 mile = 5280 ft, 1 ft = 12 in.)
- Chain the factors so units cancel. Miles → feet → inches.
- 3 mi × 5280 ft1 mi = 15840 ft Miles cancel.
- 15840 ft × 12 in1 ft = 190080 in Feet cancel, inches remain.
190,080 inches.
Area conversion
A floor measures 4 m by 3 m. Find its area in square centimetres.
- Area in m2: 4 × 3 = 12 m2 Work in the given unit first.
- 1 m2 = 100 × 100 = 10,000 cm2 Both dimensions convert, so the factor is squared.
- 12 × 10,000 = 120,000 cm2 Check: 400 cm × 300 cm = 120,000 cm2 ✓
120,000 cm2
Composite shape
An L-shape is a 10 × 8 rectangle with a 4 × 3 rectangle removed from one corner. Find its area and perimeter.
- Area: 10 × 8 = 80, minus 4 × 3 = 12 Whole shape less the bite taken out.
- Area = 68 Subtraction is faster than splitting here.
- Perimeter: the cut corner adds two edges but removes two of the same lengths. The notch does not change the total distance around an L-shape.
- P = 2(10 + 8) = 36 A surprising result worth remembering.
Area 68, perimeter 36.
Words to know
- Unit rate for conversion
- A fraction equal to 1, like 12 in1 ft.
- Metric prefixes
- kilo = 1000, centi = 1100, milli = 11000.
- Precision
- How finely a measurement is recorded — to the nearest cm, mm, and so on.
- Perimeter / area
- Distance around, in units; space inside, in square units.
Mistakes to watch for
Converting area or volume with the plain length factor. 1 m2 is 10,000 cm2, not 100 — because both the length and the width converted. For volume the factor is cubed.
Flipping the conversion fraction. Set it up so the unwanted unit appears top and bottom and cancels. If it does not cancel, the fraction is the wrong way up, and you can see that before calculating.
Getting a smaller number when converting to a smaller unit. Smaller units take more of them to cover the same thing. A shrinking number means the operation was reversed.
Dropping the units from the answer. A number without a unit cannot be checked or used, and the Mars orbiter is the expensive proof.
In 1999 NASA's $125 million Mars Climate Orbiter was lost. One team worked in pound-seconds, the other in newton-seconds.
Nobody converted. The spacecraft came in roughly 170 km too low and broke up in the Martian atmosphere.
The review board's conclusion was blunt: the failure was not the arithmetic but the missing units on the numbers being passed between teams.
Problem 1
Convert 5 feet to inches.
Worked answer
60
5 × 12 = 60 in
Problem 2
Convert 2.5 m to centimetres.
Worked answer
250
2.5 × 100 = 250 cm
Problem 3
Find the perimeter of a 7 cm by 4 cm rectangle.
Worked answer
22
2(7 + 4) = 22 cm
Problem 4
Convert 3 km to metres.
Worked answer
3000
3 × 1000 = 3000 m
Problem 5
Find the area of a 6 m by 9 m rectangle.
Worked answer
54
6 × 9 = 54 m2
Problem 6
Convert 3 miles to feet. (1 mile = 5280 ft.)
Worked answer
15840
3 × 5280 = 15,840 ft
Problem 7
Convert 4500 mm to metres.
Worked answer
4.5
4500 ÷ 1000 = 4.5 m
Problem 8
Find the area of a triangle with base 14 and perpendicular height 9.
Worked answer
63
12(14)(9) = 63
Problem 9
A trapezoid has parallel sides 6 and 10 and height 5. Find its area.
Worked answer
40
6 + 102 = 8
8 × 5 = 40
Problem 10
How many square centimetres are in 1 square metre?
Worked answer
10000
1 m2 is 100 cm × 100 cm
= 10,000 cm2
Problem 11
A rectangle has perimeter 36 and length 11. Find its width.
Worked answer
7
22 + 2w = 36
2w = 14, so w = 7
Problem 12
Convert 2.5 hours to minutes.
Worked answer
150
2.5 × 60 = 150 min
Problem 13
Convert 12 m2 to square centimetres.
Worked answer
120000
1 m2 = 10,000 cm2
12 × 10,000 = 120,000 cm2
Problem 14
Convert 3 miles to inches. (1 mi = 5280 ft, 1 ft = 12 in.)
Worked answer
190080
3 × 5280 = 15,840 ft
15,840 × 12 = 190,080 in
Problem 15
An L-shape is a 10 × 8 rectangle with a 4 × 3 rectangle cut from one corner. Find its area.
Worked answer
68
10 × 8 = 80
4 × 3 = 12
80 - 12 = 68
Problem 16
A room is 4.5 m by 3.2 m. Carpet costs $28 per square metre. Find the total cost.
Worked answer
403.20
4.5 × 3.2 = 14.4 m2
14.4 × 28 = 403.20