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E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and Volume

Converting Units for Perimeter, Area and Volume

E2.2 A1About 50 minutes
Learning goals
  • I can convert between metric units of length.
  • I can explain why 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³.
  • I can convert between units of area and between units of volume.
  • I can solve perimeter, area and volume problems that use different metric units.

Words to know

square centimetre (cm²)
The area of a square 1 cm by 1 cm.
square metre (m²)
The area of a square 1 m by 1 m. It equals 10 000 cm².
cubic metre (m³)
The volume of a cube 1 m on each side. It equals 1 000 000 cm³.
convert
To change a measurement to a different unit without changing its size, like 2 m = 200 cm.
hectare
A unit of area used for land: 10 000 m², the area of a square 100 m by 100 m.

Remember from Grade 6

In Grade 6 you converted lengths both ways: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m. To go from a bigger unit to a smaller one you multiply; to go from a smaller unit to a bigger one you divide.

Area and volume are trickier, because they use square and cubic units.

Let's start with a carpet mistake

Kai measured his bedroom floor as 3 m by 2.5 m. The carpet store sells carpet by the square metre. His sister measured the same floor in centimetres: 300 cm by 250 cm = 75 000 cm². Kai divided 75 000 by 100 and told the store he needed 750 m². That would cover a whole house! What went wrong?

Square units

1 m = 100 cm. So a square 1 m by 1 m is also 100 cm by 100 cm.

1 m² = 100 cm × 100 cm = 10 000 cm²

In the same way:

  • 1 cm² = 10 mm × 10 mm = 100 mm²
  • 1 km² = 1000 m × 1000 m = 1 000 000 m²

So Kai should have divided by 10 000: 75 000 ÷ 10 000 = 7.5 m². (Check: 3 × 2.5 = 7.5 ✓)

1 m = 100 cmeach small square:10 cm × 10 cm = 100 cm²10 × 10 = 100 small squares100 × 100 cm² = 10 000 cm²so 1 m² = 10 000 cm²
A square metre split into 10 cm by 10 cm squares (not to scale). There are 100 small squares, and each one is 100 cm², so 1 m² = 10 000 cm².

Cubic units

A cube 1 m on each side is 100 cm on each side.

1 m³ = 100 cm × 100 cm × 100 cm = 1 000 000 cm³

And 1 cm³ = 10 mm × 10 mm × 10 mm = 1000 mm³.

Length1 cm = 10 mm× 10Area1 cm² = 10 × 10= 100 mm²× 100Volume1 cm³ = 10 × 10 × 10= 1000 mm³× 1000(enlarged,not to scale)
One centimetre is 10 mm long, but a square centimetre holds 100 mm² and a cubic centimetre holds 1000 mm³. Each new dimension multiplies by 10 again.
Kind of measurementConversions
Length1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m
Area1 cm² = 100 mm², 1 m² = 10 000 cm², 1 km² = 1 000 000 m²
Volume1 cm³ = 1000 mm³, 1 m³ = 1 000 000 cm³
Capacity1 mL = 1 cm³, 1 L = 1000 mL = 1000 cm³, 1 m³ = 1000 L
Metric conversions for length, area, volume and capacity.

The safest strategy

When the measurements are in different units, convert the lengths to one unit before you calculate. Then your answer is already in the unit you want, and you never have to remember 10 000 or 1 000 000.

Example: area of a rug 250 cm by 180 cm, in m²: 2.5 m × 1.8 m = 4.5 m².

Worked example

Example 1: Area of a rug

A rug is 250 cm by 180 cm. What is its area in square metres?

  1. Method 1 (convert lengths first): 250 cm = 2.5 m and 180 cm = 1.8 m. Area = 2.5 × 1.8 = 4.5 m².
  2. Method 2 (convert the area): 250 × 180 = 45 000 cm². 45 000 ÷ 10 000 = 4.5 m².
  3. Both methods agree ✓

4.5 m²

Worked example

Example 2: Fencing a garden

A rectangular garden is 4.5 m by 320 cm. Fence panels are 2 m long. How many panels are needed to go all the way around?

  1. Convert: 320 cm = 3.2 m.
  2. Perimeter = 2 × (4.5 + 3.2) = 2 × 7.7 = 15.4 m.
  3. 15.4 ÷ 2 = 7.7 panels. You can't buy part of a panel, so round up.

8 panels (with a little left over to trim).

Worked example

Example 3: Sand for a sandbox

A sandbox is 150 cm long, 120 cm wide and is filled with sand 30 cm deep. Sand is sold by the cubic metre. How much sand is needed?

  1. Convert to metres: 1.5 m, 1.2 m and 0.3 m.
  2. Volume = 1.5 × 1.2 × 0.3 = 0.54 m³.
  3. Check: 150 × 120 × 30 = 540 000 cm³, and 540 000 ÷ 1 000 000 = 0.54 m³ ✓

0.54 m³ of sand.

Worked example

Example 4: Tiling a floor

Square tiles are 20 cm on each side. How many tiles cover a floor 3 m by 2.4 m?

  1. Convert to cm: 300 cm by 240 cm.
  2. Along the length: 300 ÷ 20 = 15 tiles. Along the width: 240 ÷ 20 = 12 tiles.
  3. 15 × 12 = 180 tiles.
  4. Check with area: 7.2 m² = 72 000 cm². One tile is 400 cm². 72 000 ÷ 400 = 180 ✓

180 tiles.

Hands-on activity

Same table, three units

You need: a tape measure, paper, a calculator if you like.

  1. Measure the top of a table in centimetres.
  2. Find its area in cm². Then convert it to m² by dividing by 10 000.
  3. Now measure it again in metres (like 1.2 m) and multiply. Do you get the same m² answer?
  4. Count how many sheets of paper (about 21.6 cm by 27.9 cm) it would take to cover the table. Estimate first, then calculate.
  5. Bonus: measure a box in cm, find its volume in cm³, then write it in m³. Why is the m³ number so small?

What to look for: both methods agree; your child explains that a m² is much bigger than a cm², so there are fewer of them.

Just for fun

Kai's mistake would have ordered 750 m² of carpet. That's enough to cover almost three tennis courts, all for one bedroom!

Your turn

1

3.5 m = ? cm

2

2 m² = ? cm²

3

45 000 cm² = ? m²

4

3 cm² = ? mm²

5

2 m³ = ? cm³

6

A table is 120 cm by 80 cm. What is its area in m²?

7

A hallway rug is 2.5 m by 90 cm. What is its perimeter in centimetres?

More practice: warm-up

1

4 m = ? cm

2

600 mm = ? cm

3

1 m² = ? cm²

4

1 cm³ = ? mm³

5

5 km = ? m

More practice: on your own

1

A patio is 4 m by 350 cm. What is its area in m²?

2

One can of paint covers 10 m². A wall is 450 cm by 260 cm. How many cans do you need for one coat?

3

A box is 40 cm by 30 cm by 25 cm. What is its volume in m³?

4

A garden bed is 6 m by 4 m. You add soil 15 cm deep. How many cubic metres of soil do you need?

5

Ravi says 3 m² = 300 cm². What mistake did he make? What is the right answer?

Challenge

1

A backyard rink is 12 m by 6 m. Before it freezes, it is flooded with water 5 cm deep. How many litres of water is that? (1 m³ = 1000 L)

2

How many 1 cm cubes fit in a 1 m cube? If you stacked them all in one tower, how tall would it be in kilometres?

3

Square tiles are 25 cm on each side. How many tiles cover a floor 4 m by 3 m? Check your answer a second way.

Check your understanding

Quiz

1.

1 m² = ?

2.

30 000 cm² = ?

3.

1 m³ = ?

4.

What is the area of a rectangle 2 m by 150 cm?

5.

2 cm² = ?

6.

A rectangle is 3 m by 40 cm. What is the best first step to find its area?

For parents and teachers

Big idea: E2.2 asks students to solve perimeter, area and volume problems that require converting from one metric unit to another. The big trap is that area and volume units do not convert by 100 like lengths do.

How to help at home: Real projects make this meaningful: paint for a wall (m²), soil for a garden (m³), tiles for a floor. Encourage converting lengths first, then calculating.

Common difficulties:

  • Converting square or cubic units like lengths (1 m² = 100 cm², wrong). Draw the 1 m square split into 10 cm squares.
  • Multiplying lengths in different units (2 m × 150 cm = 300).
  • Rounding down when you need whole items. 7.7 panels means you buy 8.

Asking "Does this answer make sense?" is an A1 skill: a bedroom is not 750 m².