E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

Converting Units Within a Measurement System

E1.3 A1About 70 minutes

Learning goals

I can convert metric units of length, mass and capacity by using powers of 10.
I can convert area units (like m² to cm²) and volume units (like cm³ to L), and explain why the factors are squared or cubed.
I can convert imperial units (inches, feet, yards, miles, ounces, pounds), including mixed units like 5 ft 3 in.
I can use conversion factors so the units cancel, and choose a sensible unit for an answer.

Words to know

conversion factor
A fraction equal to 1 that changes units, like 100 cm / 1 m or 1 ft / 12 in.

square centimetre (cm²)
The area of a square 1 cm by 1 cm.

cubic centimetre (cm³)
The volume of a cube 1 cm by 1 cm by 1 cm. It holds exactly 1 mL.

hectare (ha)
A metric unit of area for land: 10 000 m² (a square 100 m by 100 m).

capacity
How much a container can hold, often measured in mL or L.

mixed units
A measurement written with two units, like 5 ft 3 in or 2 h 15 min.

A hook: the new flooring

Ayesha's bedroom is 3 m by 4 m, so its area is 12 m². The flooring website asks for the area in cm². She types 1200, because 1 m = 100 cm. The website suggests one small box of flooring, barely enough for a doormat! What went wrong?

By the end of this lesson you will know exactly why the answer is 120 000 cm².

Length, mass and capacity in metric

In the metric system every step between neighbouring prefixes is a factor of 10. Going to a smaller unit, you get more of them, so you multiply. Going to a larger unit, you get fewer, so you divide.

The same prefixes work for metres, grams and litres: 1 kg = 1000 g, 1 L = 1000 mL.

The metric staircase. Each step down to a smaller unit multiplies the number by 10; each step up divides by 10. From km to cm is 5 steps down: × 100 000.

Conversion factors make units cancel

A conversion factor is a fraction equal to 1, like 100 cm / 1 m. Multiplying by 1 does not change the amount, only the unit. Choose the factor so that the old unit cancels:

3.75 km × (1000 m / 1 km) × (100 cm / 1 m) = 375 000 cm.

The km on top cancels the km on the bottom, and so on. This method works for every system, including imperial units and rates.

Area and volume units: square and cube the factor

A square metre is 100 cm long and 100 cm wide. So 1 m² = 100 × 100 = 10 000 cm².

A cubic metre is 100 cm long, wide and high. So 1 m³ = 100 × 100 × 100 = 1 000 000 cm³.

The link between volume and capacity: 1 cm³ = 1 mL, so 1000 cm³ = 1 L, and 1 m³ = 1000 L.

Why 1 m² = 10 000 cm²: the square is 100 cm on each side, so it holds 100 rows of 100 cm² squares. (The grid shows 10 cm squares to keep it readable.)

A 10 cm cube has a volume of 1000 cm³, and it holds exactly 1 litre. That is why a litre is also called a cubic decimetre (1 dm³).

Quantity	Metric	Imperial
Length	10 mm = 1 cm, 100 cm = 1 m, 1000 m = 1 km	12 in = 1 ft, 3 ft = 1 yd, 1760 yd = 1 mi (5280 ft = 1 mi)
Area	1 m² = 10 000 cm², 1 ha = 10 000 m², 1 km² = 100 ha	1 ft² = 144 in², 1 yd² = 9 ft², 1 acre = 43 560 ft²
Volume and capacity	1 cm³ = 1 mL, 1000 mL = 1 L, 1 m³ = 1000 L	1 ft³ = 1728 in³, 1 yd³ = 27 ft³
Mass	1000 mg = 1 g, 1000 g = 1 kg, 1000 kg = 1 t	16 oz = 1 lb

Relationships within each system. The imperial numbers (12, 3, 1760, 16) have to be memorized or looked up; metric ones are all powers of 10.

Worked example

Example 1: Ayesha's floor

Convert 12 m² to cm².

1 m² = 100 cm × 100 cm = 10 000 cm².
12 m² × (10 000 cm² / 1 m²) = 120 000 cm².
Check with dimensions: 3 m = 300 cm and 4 m = 400 cm, so 300 × 400 = 120 000 cm². ✓

120 000 cm². Ayesha multiplied by 100 instead of 100² = 10 000.

Worked example

Example 2: Aquarium

An aquarium is 60 cm × 30 cm × 35 cm. How many litres of water can it hold?

V = 60 × 30 × 35 = 63 000 cm³.
1 cm³ = 1 mL, so that is 63 000 mL.
63 000 mL ÷ 1000 = 63 L.

63 L

Worked example

Example 3: Imperial area

A deck is 12 ft by 15 ft. Find its area in square feet and in square yards.

A = 12 × 15 = 180 ft².
1 yd² = 3 ft × 3 ft = 9 ft², so divide by 9: 180 ÷ 9 = 20 yd².
Check: 12 ft = 4 yd and 15 ft = 5 yd, and 4 × 5 = 20 yd². ✓

180 ft² = 20 yd²

Worked example

Example 4: Mixed units

A curtain rod is 150 in long. Write the length in feet and inches.

150 ÷ 12 = 12 remainder 6.
So 150 in = 12 ft 6 in.
In decimal feet that is 12.5 ft, because 6 in is half of a foot. (Not 12.6 ft!)

12 ft 6 in (or 12.5 ft)

Worked example

Example 5: A rate

A car travels at 90 km/h. How many metres does it travel each second?

90 km/h = 90 km per 1 h.
90 km × (1000 m / 1 km) = 90 000 m, and 1 h × (3600 s / 1 h) = 3600 s.
90 000 m ÷ 3600 s = 25 m/s.

25 m/s

Watch out!

For area, square the length factor: 1 m² = 100² cm². For volume, cube it: 1 m³ = 100³ cm³.
In mixed units, the part after the unit is not a decimal: 5 ft 6 in = 5.5 ft, not 5.6 ft.
Estimate first. A smaller unit means a bigger number. If you convert m² to cm² and get a smaller number, something is wrong.

Hands-on activity

Kitchen and bedroom conversions

You need: a measuring tape, a calculator, a box (cereal or shoe box), and a container with a capacity label (like a 2 L juice jug).

Measure the inside of the box in cm. Find its volume in cm³ and convert to mL and L.
Fill the box partway with dry rice or pasta using a measuring cup to check if your answer is reasonable. (Pour it back afterwards.)
Measure a room in metres. Find its area in m², then convert to cm² and to hectares. Which unit is sensible for a room? (m².)
Find your height in inches with the tape's inch side. Write it in feet and inches.

Model answer: a box 19 cm × 7 cm × 28 cm has V = 3724 cm³ = 3724 mL ≈ 3.7 L.

Just for fun

A litre and a cubic centimetre went on a road trip. The litre needed 1000 times more trunk space.

Your turn

Convert 4.6 m to millimetres.

Convert 2500 mL to litres.

Convert 2.5 m² to cm².

A box has a volume of 38 000 cm³. How many litres does it hold?

Write 7 ft 4 in in inches.

Convert 2 yd² to square feet.

How many 250 mL glasses can you fill from a 1.5 L bottle?

More practice: warm-up

3 km = ? m

450 cm = ? m

2 ft = ? in

3 lb = ? oz

1 L = ? mL

More practice: on your own

A soccer field is 120 m by 80 m. What is its area in hectares?

A pool holds 375 m³ of water. How many litres is that?

How many cubic inches are in 3 ft³?

Convert 100 km/h to m/s, to one decimal place.

A board is 2.4 m long. How many 30 cm pieces can be cut from it (ignore the saw cut)?

Challenge

A phone screen is 7 cm by 15 cm. Find its area in cm² and in m².

A garden is 30 ft by 40 ft. You want topsoil 6 in deep. How many cubic feet is that? How many cubic yards (to one decimal)?

25 mm of rain falls on a flat roof that is 10 m by 12 m. How many litres of water land on the roof?

Check your understanding

Quiz

1 m² is equal to:

How much does a 1 cm³ space hold?

3 ft 8 in is equal to:

4.2 L is equal to:

Which conversion factor changes yards into feet?

A tank holds 2 m³ of water. How many litres is that?

For parents and teachers

Big idea: Expectation E1.3 asks students to solve problems with different units within a measurement system and between systems. This lesson covers within-system conversions; the next lesson covers between systems.

Common difficulties:

Area and volume conversions. The most common error is converting m² to cm² by × 100. Draw the square and ask how many rows and columns of cm² fit.
Mixed units as decimals (5 ft 6 in written as 5.6 ft).
Multiplying when they should divide. Ask: "Is the new unit smaller or larger? Should the number get bigger or smaller?"

How to help: Cooking and home projects are full of conversions: millilitres to litres, grams to kilograms, inches to feet. Let your teen calculate how much paint, soil or flooring a project needs.