Converting Units of Area and Volume

See why $1\text{ m}^{2} = 10\,000\text{ cm}^{2}$ and $1\text{ m}^{3} = 1\,000\,000\text{ cm}^{3} = 1000$ L, change areas, volumes and capacities from one unit to another, and use them for tiles, boxes, tanks and costs.

Before you start

$1\text{ m} = 100\text{ cm}$ and $1\text{ cm} = 10\text{ mm}$ . So $2.4\text{ m} = 240\text{ cm}$ , $80\text{ cm} = 0.8\text{ m}$ and $5\text{ mm} = 0.5\text{ cm}$ .
The area of a rectangle is length $\times$ breadth; the volume of a cuboid is length $\times$ breadth $\times$ height. A square of side 1 cm has an area of $1\text{ cm}^{2}$ ; a cube of edge 1 cm has a volume of $1\text{ cm}^{3}$ .
$1\text{ L} = 1000\text{ mL}$ , and $1\text{ cm}^{3}$ of space holds $1\text{ mL}$ of water.

Units of Area: Square Metres and Square Centimetres

An area is measured by counting squares. A square centimetre ( $1\text{ cm}^{2}$ ) is the area of a square of side 1 cm. A square metre ( $1\text{ m}^{2}$ ) is the area of a square of side 1 m.

Because $1\text{ m} = 100\text{ cm}$ , it is tempting to think that $1\text{ m}^{2} = 100\text{ cm}^{2}$ . It is not. A square of side 1 m is 100 cm long and 100 cm wide, so it holds 100 rows of 100 small squares of side 1 cm.

The square is ruled into blocks of side 10 cm: 10 rows of 10 blocks, 100 blocks in all. Each block holds $10 \times 10 = 100$ squares of side 1 cm. So the whole square holds $100 \times 100 = 10\,000$ squares of side 1 cm.

Units of area

$1\text{ m}^{2} = 100\text{ cm} \times 100\text{ cm} = 10\,000\text{ cm}^{2}$

To change $\text{m}^{2}$ to $\text{cm}^{2}$ , multiply by $10\,000$ .

To change $\text{cm}^{2}$ to $\text{m}^{2}$ , divide by $10\,000$ .

A quick check: the smaller unit always gives the bigger number. Multiplying by $10\,000$ moves the decimal point 4 places to the right; dividing by $10\,000$ moves it 4 places to the left.

Example 1: Changing units of area

Change (a) $2.5\text{ m}^{2}$ to $\text{cm}^{2}$ , (b) $0.064\text{ m}^{2}$ to $\text{cm}^{2}$ , (c) $4800\text{ cm}^{2}$ to $\text{m}^{2}$ , (d) $25\text{ cm}^{2}$ to $\text{m}^{2}$ .

Solution

From $\text{m}^{2}$ to $\text{cm}^{2}$ (a smaller unit, so a bigger number): multiply by $10\,000$ .

(a) $2.5\text{ m}^{2} = 2.5 \times 10\,000\text{ cm}^{2} = 25\,000\text{ cm}^{2}$

(b) $0.064\text{ m}^{2} = 0.064 \times 10\,000\text{ cm}^{2} = 640\text{ cm}^{2}$

From $\text{cm}^{2}$ to $\text{m}^{2}$ (a bigger unit, so a smaller number): divide by $10\,000$ .

(d) $25\text{ cm}^{2} = 25 \div 10\,000\text{ m}^{2} = 0.0025\text{ m}^{2}$

(a) $25\,000\text{ cm}^{2}$ (b) $640\text{ cm}^{2}$ (c) $0.48\text{ m}^{2}$ (d) $0.0025\text{ m}^{2}$

Lengths in Different Units: Change the Lengths First

When the sides of a figure are given in different units, change them to one unit before you multiply. Multiplying $2.4$ by $80$ gives $192$ , which is neither the area in $\text{cm}^{2}$ nor the area in $\text{m}^{2}$ .

Choose the unit the answer asks for: change every length to centimetres for an answer in $\text{cm}^{2}$ , or to metres for an answer in $\text{m}^{2}$ . This safe method needs no conversion of areas at all.

The safe method

Change all the lengths to one unit first, then multiply.

Lengths in cm give an area in $\text{cm}^{2}$ ; lengths in m give an area in $\text{m}^{2}$ .

Example 2: A rectangle measured in metres and centimetres

A rectangular mat is 2.4 m long and 80 cm wide. Find its area (a) in $\text{cm}^{2}$ , (b) in $\text{m}^{2}$ .

Solution

(a) In centimetres: $2.4\text{ m} = 240\text{ cm}$ .

Area $= 240 \times 80$

$= 19\,200\text{ cm}^{2}$

(b) In metres: $80\text{ cm} = 0.8\text{ m}$ .

Area $= 2.4 \times 0.8$

$= 1.92\text{ m}^{2}$

Check: $19\,200 \div 10\,000 = 1.92$ , so the two answers agree.

The area of the mat is $19\,200\text{ cm}^{2}$ , which is $1.92\text{ m}^{2}$ .

Example 3: A triangle, with the answer in a given unit

A triangular piece of cloth for a class banner has a base of 1.5 m and a perpendicular height of 60 cm. Find its area in $\text{cm}^{2}$ , then write it in $\text{m}^{2}$ .

Solution

The answer is wanted in $\text{cm}^{2}$ , so change the base to centimetres: $1.5\text{ m} = 150\text{ cm}$ .

Area $= \frac{1}{2} \times$ base $\times$ height

$= \frac{1}{2} \times 150 \times 60$

$= 4500\text{ cm}^{2}$

In $\text{m}^{2}$ : $4500 \div 10\,000 = 0.45\text{ m}^{2}$ .

Check in metres: $\frac{1}{2} \times 1.5 \times 0.6 = 0.45\text{ m}^{2}$ .

The area of the cloth is $4500\text{ cm}^{2}$ , which is $0.45\text{ m}^{2}$ .

Units of Volume: Cubic Metres and Cubic Centimetres

A cubic centimetre ( $1\text{ cm}^{3}$ ) is the volume of a cube of edge 1 cm. A cubic metre ( $1\text{ m}^{3}$ ) is the volume of a cube of edge 1 m.

A cube of edge 1 m is 100 cm long, 100 cm wide and 100 cm high. So its volume is $100 \times 100 \times 100 = 1\,000\,000\text{ cm}^{3}$ : not 100, and not 1000.

Each square on the faces is 10 cm by 10 cm: it is the face of a small cube of edge 10 cm, like the shaded one, with a volume of $10 \times 10 \times 10 = 1000\text{ cm}^{3}$ . The big cube holds 10 layers, each of 10 rows of 10 small cubes: $1000$ small cubes. So its volume is $1000 \times 1000 = 1\,000\,000\text{ cm}^{3}$ .

Units of volume

$1\text{ m}^{3} = 100\text{ cm} \times 100\text{ cm} \times 100\text{ cm} = 1\,000\,000\text{ cm}^{3}$

To change $\text{m}^{3}$ to $\text{cm}^{3}$ , multiply by $1\,000\,000$ .

To change $\text{cm}^{3}$ to $\text{m}^{3}$ , divide by $1\,000\,000$ .

Multiplying by $1\,000\,000$ moves the decimal point 6 places to the right; dividing moves it 6 places to the left. Here too, the safe method works: change the lengths to one unit first, then multiply.

Example 4: Changing units of volume

Change (a) $3\text{ m}^{3}$ to $\text{cm}^{3}$ , (b) $0.45\text{ m}^{3}$ to $\text{cm}^{3}$ , (c) $750\,000\text{ cm}^{3}$ to $\text{m}^{3}$ , (d) $2000\text{ cm}^{3}$ to $\text{m}^{3}$ .

Solution

From $\text{m}^{3}$ to $\text{cm}^{3}$ : multiply by $1\,000\,000$ .

(a) $3\text{ m}^{3} = 3 \times 1\,000\,000\text{ cm}^{3} = 3\,000\,000\text{ cm}^{3}$

(b) $0.45\text{ m}^{3} = 0.45 \times 1\,000\,000\text{ cm}^{3} = 450\,000\text{ cm}^{3}$

From $\text{cm}^{3}$ to $\text{m}^{3}$ : divide by $1\,000\,000$ .

(d) $2000\text{ cm}^{3} = 2000 \div 1\,000\,000\text{ m}^{3} = 0.002\text{ m}^{3}$

(a) $3\,000\,000\text{ cm}^{3}$ (b) $450\,000\text{ cm}^{3}$ (c) $0.75\text{ m}^{3}$ (d) $0.002\text{ m}^{3}$

Example 5: Concrete for a step

Mr Tan builds a concrete step in the shape of a cuboid, 1.2 m long, 40 cm deep and 15 cm high. How much concrete does he need? Give your answer in $\text{m}^{3}$ and in $\text{cm}^{3}$ .

Solution

In metres: $40\text{ cm} = 0.4\text{ m}$ and $15\text{ cm} = 0.15\text{ m}$ .

Volume $= 1.2 \times 0.4 \times 0.15$

$= 0.072\text{ m}^{3}$

In centimetres: $1.2\text{ m} = 120\text{ cm}$ .

Volume $= 120 \times 40 \times 15$

$= 72\,000\text{ cm}^{3}$

Check: $72\,000 \div 1\,000\,000 = 0.072$ , so the two answers agree.

Mr Tan needs $0.072\text{ m}^{3}$ of concrete, which is $72\,000\text{ cm}^{3}$ .

Capacity: Litres and Cubic Metres

The capacity of a container is the volume of the space inside it: how much it can hold. Capacity is usually given in litres (L) and millilitres (mL).

A space of $1\text{ cm}^{3}$ holds $1\text{ mL}$ , so a cube of edge 10 cm, with a volume of $1000\text{ cm}^{3}$ , holds exactly 1 L. A cube of edge 1 m holds $1\,000\,000\text{ cm}^{3}$ , which is $1\,000\,000\text{ mL}$ , and $1\,000\,000 \div 1000 = 1000$ L.

A container has walls, so it is larger outside than inside. Its capacity is worked out from its inside (internal) measurements.

Units of capacity

$1\text{ cm}^{3} = 1\text{ mL}$ and $1000\text{ cm}^{3} = 1\text{ L}$

$1\text{ m}^{3} = 1\,000\,000\text{ cm}^{3} = 1000\text{ L}$

The capacity of a container is found from its inside measurements.

Example 6: Changing units of capacity

Change (a) $2.5\text{ m}^{3}$ to litres, (b) 600 L to $\text{m}^{3}$ , (c) $45\,000\text{ cm}^{3}$ to litres, (d) $0.004\text{ m}^{3}$ to millilitres.

Solution

(a) $1\text{ m}^{3} = 1000$ L, so $2.5\text{ m}^{3} = 2.5 \times 1000\text{ L} = 2500$ L.

(b) $600\text{ L} = 600 \div 1000\text{ m}^{3} = 0.6\text{ m}^{3}$

(d) First to $\text{cm}^{3}$ : $0.004\text{ m}^{3} = 0.004 \times 1\,000\,000\text{ cm}^{3} = 4000\text{ cm}^{3}$ .

Since $1\text{ cm}^{3} = 1\text{ mL}$ , this is $4000\text{ mL}$ (that is, 4 L).

(a) 2500 L (b) $0.6\text{ m}^{3}$ (c) 45 L (d) 4000 mL

Example 7: A water tank on a school roof

A water tank on the roof of a school canteen is a cuboid. Inside, it is 2.4 m long, 1.5 m wide and 1.2 m high.

(a) Find its capacity in $\text{m}^{3}$ and in litres.

(b) The water in it is 90 cm deep. How many litres of water are in the tank?

Solution

(a) The measurements given are inside measurements, so they give the capacity.

Capacity $= 2.4 \times 1.5 \times 1.2$

$= 4.32\text{ m}^{3}$

In litres: $4.32 \times 1000 = 4320$ L.

(b) The water is a cuboid 2.4 m long, 1.5 m wide and $90\text{ cm} = 0.9\text{ m}$ deep.

Volume of water $= 2.4 \times 1.5 \times 0.9$

$= 3.24\text{ m}^{3}$

In litres: $3.24 \times 1000 = 3240$ L.

(a) The capacity is $4.32\text{ m}^{3}$ , or 4320 L. (b) There are 3240 L of water in the tank.

Comparing Quantities in Different Units

To compare or order areas, volumes or capacities given in different units, change them all to one unit first. The smaller unit is usually easier, since it avoids long decimals.

Example 8: Ordering areas

Arrange in order of size, starting with the smallest: $0.03\text{ m}^{2}$ , $280\text{ cm}^{2}$ , $0.0025\text{ m}^{2}$ .

Solution

Change each to $\text{cm}^{2}$ by multiplying the areas in $\text{m}^{2}$ by $10\,000$ :

$0.03\text{ m}^{2} = 0.03 \times 10\,000\text{ cm}^{2} = 300\text{ cm}^{2}$

$280\text{ cm}^{2}$ stays as it is.

$0.0025\text{ m}^{2} = 0.0025 \times 10\,000\text{ cm}^{2} = 25\text{ cm}^{2}$

In order: $25 < 280 < 300$ .

Smallest first: $0.0025\text{ m}^{2}$ , $280\text{ cm}^{2}$ , $0.03\text{ m}^{2}$ .

Tiles, Boxes and Costs

To find how many whole tiles fit on a floor without cutting any, or how many boxes fit in a carton, count how many fit along each edge, then multiply. Work with the lengths in one unit.

Dividing the whole area by the area of one tile (or the whole volume by the volume of one box) gives the same answer only when they fit exactly. When they do not, the strips left over at the edges are too narrow for a whole tile or box, and dividing the areas or volumes gives too many.

When the tiles do not fit exactly, whole tiles alone leave a strip of the floor bare. If tiles may be cut so that the whole floor is covered, each space at the edge takes one more tile, cut to fit: the number along each edge is then rounded up.

Counting tiles and boxes

Number along an edge $=$ length of the edge $\div$ length of the tile (or box) along that edge.

For whole tiles that fit without cutting, or boxes that fit, round this down to a whole number, since part of a tile or box does not fit.

For tiles cut to cover the whole floor, round it up, since the strip left at the edge still needs a tile (each cut tile filling one space, its offcut not used).

Number of tiles $=$ the two numbers along the edges multiplied. Number of boxes $=$ the three numbers along the edges multiplied.

Example 9: Tiles that fit exactly

A kitchen floor is a rectangle 3 m by 2.4 m. It is covered with square tiles of side 20 cm. The tiles are sold in boxes of 25 at $18.50 a box.

(a) How many tiles are needed? (b) How many boxes must be bought, and what do they cost?

Solution

(a) In centimetres the floor is 300 cm by 240 cm.

Along the 300 cm side: $300 \div 20 = 15$ tiles.

Along the 240 cm side: $240 \div 20 = 12$ tiles.

Number of tiles $= 15 \times 12 = 180$

Here the tiles fit exactly, so dividing the areas agrees: $300 \times 240 = 72\,000\text{ cm}^{2}$ , one tile is $20 \times 20 = 400\text{ cm}^{2}$ , and $72\,000 \div 400 = 180$ .

(b) $180 \div 25 = 7.2$ . Seven boxes hold only 175 tiles, so 8 boxes must be bought (round up: a part of a box cannot be bought).

Cost $= 8 \times 18.50 = 148$

(a) 180 tiles are needed. (b) 8 boxes must be bought, costing $148.

Example 10: Cubes that do not fit exactly

A box is 10 cm long, 7 cm wide and 5 cm high inside. How many cubes of edge 2 cm can be packed into it?

Solution

Along the 10 cm length: $10 \div 2 = 5$ cubes.

Along the 7 cm width: $7 \div 2 = 3.5$ , so 3 cubes, with 1 cm left over.

Up the 5 cm height: $5 \div 2 = 2.5$ , so 2 cubes, with 1 cm left over.

Number of cubes $= 5 \times 3 \times 2 = 30$

Dividing the volumes would give $10 \times 7 \times 5 = 350\text{ cm}^{3}$ , one cube $2 \times 2 \times 2 = 8\text{ cm}^{3}$ , and $350 \div 8 = 43.75$ . That is wrong: the gaps 1 cm wide at the back and 1 cm high at the top cannot hold a cube of edge 2 cm.

30 cubes can be packed into the box.

Example 11: Carpet priced per square metre

Siti's bedroom floor is a rectangle 450 cm by 380 cm. Carpet costs $32.50 per $\text{m}^{2}$ . Find the cost of carpeting the whole floor.

Solution

The price is per $\text{m}^{2}$ , so work in metres: $450\text{ cm} = 4.5\text{ m}$ and $380\text{ cm} = 3.8\text{ m}$ .

Area $= 4.5 \times 3.8$

$= 17.1\text{ m}^{2}$

Check: $450 \times 380 = 171\,000\text{ cm}^{2}$ , and $171\,000 \div 10\,000 = 17.1\text{ m}^{2}$ .

Cost $= 17.1 \times 32.50 = 555.75$

Carpeting the floor costs $555.75.

Common Mistakes

Changing $\text{m}^{2}$ to $\text{cm}^{2}$ by multiplying by 100. A square metre is 100 cm by 100 cm, so $1\text{ m}^{2} = 10\,000\text{ cm}^{2}$ , not $100\text{ cm}^{2}$ .
Changing $\text{m}^{3}$ to $\text{cm}^{3}$ by multiplying by 100 or by 1000. A cubic metre is 100 cm by 100 cm by 100 cm, so $1\text{ m}^{3} = 1\,000\,000\text{ cm}^{3}$ .
Mixing up $1\text{ m}^{3} = 1000$ L with $1\text{ m}^{3} = 1000\text{ cm}^{3}$ . It is 1000 litres, and each litre is $1000\text{ cm}^{3}$ .
Moving the decimal point the wrong way. Changing to a smaller unit ( $\text{m}^{2}$ to $\text{cm}^{2}$ ) gives a bigger number; changing to a bigger unit gives a smaller number.
Multiplying lengths in different units: $2.4\text{ m} \times 80\text{ cm}$ is not $192$ of anything. Change the lengths to one unit first.
Counting tiles or boxes by dividing areas or volumes when they do not fit exactly. Count along each edge, then multiply: round down for whole tiles or boxes that fit, round up for tiles cut to cover the whole floor.
Working out the capacity of a container from its outside measurements. Capacity uses the inside measurements.

Practice

Try each question before you open its solution. Give answers that are not exact correct to 3 significant figures.

Practice 1: Changing units of area

Change (a) $7\text{ m}^{2}$ to $\text{cm}^{2}$ , (b) $0.45\text{ m}^{2}$ to $\text{cm}^{2}$ , (c) $36\,000\text{ cm}^{2}$ to $\text{m}^{2}$ , (d) $90\text{ cm}^{2}$ to $\text{m}^{2}$ .

ShowHide solution

(a) $7 \times 10\,000 = 70\,000$ , so $7\text{ m}^{2} = 70\,000\text{ cm}^{2}$ .

(b) $0.45 \times 10\,000 = 4500$ , so $0.45\text{ m}^{2} = 4500\text{ cm}^{2}$ .

(d) $90 \div 10\,000 = 0.009$ , so $90\text{ cm}^{2} = 0.009\text{ m}^{2}$ .

(a) $70\,000\text{ cm}^{2}$ (b) $4500\text{ cm}^{2}$ (c) $3.6\text{ m}^{2}$ (d) $0.009\text{ m}^{2}$

Practice 2: Changing units of volume and capacity

Change (a) $0.6\text{ m}^{3}$ to $\text{cm}^{3}$ , (b) $2\,400\,000\text{ cm}^{3}$ to $\text{m}^{3}$ , (c) $8500\text{ cm}^{3}$ to $\text{m}^{3}$ , (d) $1.25\text{ m}^{3}$ to litres, (e) 350 L to $\text{m}^{3}$ .

ShowHide solution

(a) $0.6 \times 1\,000\,000 = 600\,000$ , so $0.6\text{ m}^{3} = 600\,000\text{ cm}^{3}$ .

(b) $2\,400\,000 \div 1\,000\,000 = 2.4$ , so $2\,400\,000\text{ cm}^{3} = 2.4\text{ m}^{3}$ .

(d) $1\text{ m}^{3} = 1000$ L, so $1.25\text{ m}^{3} = 1.25 \times 1000\text{ L} = 1250$ L.

(e) $350 \div 1000 = 0.35$ , so $350\text{ L} = 0.35\text{ m}^{3}$ .

(a) $600\,000\text{ cm}^{3}$ (b) $2.4\text{ m}^{3}$ (c) $0.0085\text{ m}^{3}$ (d) 1250 L (e) $0.35\text{ m}^{3}$

Practice 3: True or false?

Say whether each statement is true or false. Correct each one that is false.

(a) $1\text{ m}^{2} = 100\text{ cm}^{2}$ (b) $1\text{ m}^{3} = 1000$ L (c) $5\text{ m}^{3} = 5000\text{ cm}^{3}$ (d) $0.2\text{ m}^{2} = 2000\text{ cm}^{2}$ (e) $30\,000\text{ cm}^{3} = 0.3\text{ m}^{3}$

ShowHide solution

(a) False. A square metre is 100 cm by 100 cm: $1\text{ m}^{2} = 10\,000\text{ cm}^{2}$ .

(b) True. $1\text{ m}^{3} = 1\,000\,000\text{ cm}^{3}$ and $1\,000\,000 \div 1000 = 1000$ L.

(c) False. $5 \times 1\,000\,000 = 5\,000\,000$ : $5\text{ m}^{3} = 5\,000\,000\text{ cm}^{3}$ . (It is $5000$ litres.)

(d) True. $0.2 \times 10\,000 = 2000$ .

(e) False. $30\,000 \div 1\,000\,000 = 0.03$ : $30\,000\text{ cm}^{3} = 0.03\text{ m}^{3}$ .

(a) False: $10\,000\text{ cm}^{2}$ (b) True (c) False: $5\,000\,000\text{ cm}^{3}$ (d) True (e) False: $0.03\text{ m}^{3}$

Practice 4: Ordering volumes

Arrange in order of size, starting with the smallest: $0.15\text{ m}^{3}$ , $140\,000\text{ cm}^{3}$ , 155 L, $0.016\text{ m}^{3}$ .

ShowHide solution

Change each to $\text{cm}^{3}$ .

$0.15\text{ m}^{3} = 0.15 \times 1\,000\,000 = 150\,000\text{ cm}^{3}$

$140\,000\text{ cm}^{3}$ stays as it is.

$155\text{ L} = 155 \times 1000 = 155\,000\text{ cm}^{3}$

$0.016\text{ m}^{3} = 0.016 \times 1\,000\,000 = 16\,000\text{ cm}^{3}$

In order: $16\,000 < 140\,000 < 150\,000 < 155\,000$ .

Smallest first: $0.016\text{ m}^{3}$ , $140\,000\text{ cm}^{3}$ , $0.15\text{ m}^{3}$ , 155 L.

Practice 5: A notice board

A rectangular notice board in a classroom is 1.6 m long and 45 cm high. Find its area (a) in $\text{cm}^{2}$ , (b) in $\text{m}^{2}$ .

ShowHide solution

(a) $1.6\text{ m} = 160\text{ cm}$ . Area $= 160 \times 45 = 7200\text{ cm}^{2}$ .

(b) $45\text{ cm} = 0.45\text{ m}$ . Area $= 1.6 \times 0.45 = 0.72\text{ m}^{2}$ .

Check: $7200 \div 10\,000 = 0.72$ .

The area is $7200\text{ cm}^{2}$ , which is $0.72\text{ m}^{2}$ .

Practice 6: A flower bed shaped like a trapezium

A flower bed is a trapezium with parallel sides 1.2 m and 80 cm, which are 50 cm apart. Find its area in $\text{m}^{2}$ and in $\text{cm}^{2}$ .

ShowHide solution

In metres: $80\text{ cm} = 0.8\text{ m}$ and $50\text{ cm} = 0.5\text{ m}$ .

Area $= \frac{1}{2} \times (a + b) \times h$

$= \frac{1}{2} \times (1.2 + 0.8) \times 0.5$

$= 0.5\text{ m}^{2}$

In $\text{cm}^{2}$ : $0.5 \times 10\,000 = 5000\text{ cm}^{2}$ .

Check in centimetres: $\frac{1}{2} \times (120 + 80) \times 50 = 100 \times 50 = 5000\text{ cm}^{2}$ .

The area of the flower bed is $0.5\text{ m}^{2}$ , which is $5000\text{ cm}^{2}$ .

Practice 7: A shelf board measured in millimetres

Ravi's shelf is a wooden board 1.8 m long, 25 cm wide and 20 mm thick. Find the volume of wood in the shelf (a) in $\text{cm}^{3}$ , (b) in $\text{m}^{3}$ .

ShowHide solution

(a) Change every length to centimetres: $1.8\text{ m} = 180\text{ cm}$ and $20\text{ mm} = 2\text{ cm}$ .

Volume $= 180 \times 25 \times 2$

$= 9000\text{ cm}^{3}$

(b) $9000 \div 1\,000\,000 = 0.009$ , so the volume is $0.009\text{ m}^{3}$ .

Check in metres: $1.8 \times 0.25 \times 0.02 = 0.009\text{ m}^{3}$ .

The shelf has $9000\text{ cm}^{3}$ of wood, which is $0.009\text{ m}^{3}$ .

Practice 8: Mei Ling's fish tank

Mei Ling's fish tank has no lid. Outside, it is 58 cm long, 30 cm wide and 33 cm high. The glass is 1 cm thick, so inside it is 56 cm long, 28 cm wide and 32 cm deep, as drawn below.

(a) Find the capacity of the tank in litres. (b) She fills it with water to a depth of 25 cm. How many litres of water are in the tank?

ShowHide solution

(a) The capacity uses the inside measurements: 56 cm, 28 cm and 32 cm.

Capacity $= 56 \times 28 \times 32$

$= 50\,176\text{ cm}^{3}$

$50\,176 \div 1000 = 50.176$ , so the capacity is $50.176$ L (an exact value, so it is not rounded).

(b) The water fills the inside to a depth of 25 cm.

Volume of water $= 56 \times 28 \times 25$

$= 39\,200\text{ cm}^{3}$

$39\,200 \div 1000 = 39.2$ L

(a) The capacity is 50.176 L. (b) There are 39.2 L of water in the tank.

Practice 9: Cartons in a van

The space at the back of a delivery van is a cuboid 2.5 m long, 1.4 m wide and 1.1 m high. Cartons 40 cm long, 30 cm wide and 25 cm high are packed in, all the same way up, with their 40 cm edges along the length of the van and their 30 cm edges across its width.

(a) How many cartons fit? (b) Explain why $\text{volume of the space} \div \text{volume of a carton}$ gives the wrong answer.

ShowHide solution

(a) In centimetres the space is 250 cm long, 140 cm wide and 110 cm high.

Along the length: $250 \div 40 = 6.25$ , so 6 cartons.

Across the width: $140 \div 30 = 4.66\ldots$ , so 4 cartons.

Up the height: $110 \div 25 = 4.4$ , so 4 cartons.

Number of cartons $= 6 \times 4 \times 4 = 96$

(b) Volume of the space $= 250 \times 140 \times 110 = 3\,850\,000\text{ cm}^{3}$ (that is, $3.85\text{ m}^{3}$ ). Volume of a carton $= 40 \times 30 \times 25 = 30\,000\text{ cm}^{3}$ . $3\,850\,000 \div 30\,000 = 128.3\ldots$

This is too many, because the cartons do not fit exactly: 10 cm is left along the length, 20 cm across the width and 10 cm at the top, and no carton fits in those gaps.

(a) 96 cartons fit. (b) The cartons do not fit exactly, so the gaps left at the edges are counted as if they held cartons.

Practice 10: Tiles on a wall, whole and cut

Students of a school art club fix square tiles of side 40 cm on a wall 2.1 m wide and 1.5 m high. They start at the bottom left corner and cut no tiles.

(a) How many whole tiles can they fix? (b) What area of the wall, in $\text{m}^{2}$ , is left uncovered?

(c) Next term the club tiles a second wall of the same size, and this time cuts tiles so that the whole wall is covered, each cut tile filling one space at the edge. How many tiles do they need?

ShowHide solution

(a) In centimetres the wall is 210 cm by 150 cm.

Across: $210 \div 40 = 5.25$ , so 5 tiles.

Up: $150 \div 40 = 3.75$ , so 3 tiles.

Number of tiles $= 5 \times 3 = 15$

(Dividing the areas, $31\,500 \div 1600 \approx 19.7$ , would be wrong: the tiles do not fit exactly.)

(b) Area of the wall $= 2.1 \times 1.5 = 3.15\text{ m}^{2}$ .

One tile: $40\text{ cm} = 0.4\text{ m}$ , so its area is $0.4 \times 0.4 = 0.16\text{ m}^{2}$ .

Area of 15 tiles $= 15 \times 0.16 = 2.4\text{ m}^{2}$

Area uncovered $= 3.15 - 2.4 = 0.75\text{ m}^{2}$

Check in $\text{cm}^{2}$ : $31\,500 - 15 \times 1600 = 31\,500 - 24\,000 = 7500\text{ cm}^{2}$ , and $7500 \div 10\,000 = 0.75$ .

Across: $210 \div 40 = 5.25$ , so 5 whole tiles and one more cut to fit: 6 tiles.

Up: $150 \div 40 = 3.75$ , so 3 whole tiles and one more cut to fit: 4 tiles.

Number of tiles $= 6 \times 4 = 24$

Check: 24 tiles cover $24 \times 0.16 = 3.84\text{ m}^{2}$ , more than the wall's $3.15\text{ m}^{2}$ , as they must, since the cut-off pieces are not used.

(a) 15 whole tiles. (b) $0.75\text{ m}^{2}$ of the wall is left uncovered. (c) 24 tiles are needed.

Practice 11: Soil for a planter box

Hui Min's class fills a planter box in the school garden with soil. The soil inside it is a cuboid 2.4 m long, 90 cm wide and 40 cm deep. Soil is sold in bags of 50 L.

(a) Find the volume of soil in $\text{m}^{3}$ and in litres. (b) How many bags of soil must the class buy?

ShowHide solution

(a) In metres: $90\text{ cm} = 0.9\text{ m}$ and $40\text{ cm} = 0.4\text{ m}$ .

Volume $= 2.4 \times 0.9 \times 0.4$

$= 0.864\text{ m}^{3}$

In litres: $0.864 \times 1000 = 864$ L.

(b) $864 \div 50 = 17.28$ . Seventeen bags hold only 850 L, which is not enough, so round up: the class must buy 18 bags.

(a) $0.864\text{ m}^{3}$ , which is 864 L. (b) The class must buy 18 bags.

Practice 12: The depth of the water

A cuboid tank is 60 cm high, and its inside base is 1.2 m long and 80 cm wide. Mr Tan pours 384 L of water into the empty tank. Find the depth of the water, $h$ cm.

(not drawn to scale)

ShowHide solution

The depth is wanted in cm, so work in centimetres. $1.2\text{ m} = 120\text{ cm}$ , and $384\text{ L} = 384 \times 1000 = 384\,000\text{ cm}^{3}$ .

The water is a cuboid with the tank's inside base, so its volume is length $\times$ breadth $\times$ depth.

$120 \times 80 \times h = 384\,000$

$9600h = 384\,000$

$h = 40$ (divide both sides by 9600)

Check: $120 \times 80 \times 40 = 384\,000\text{ cm}^{3} = 384$ L, and 40 cm is less than the tank's 60 cm, so the water does not overflow.

The water is 40 cm deep.

Key Points

$1\text{ m}^{2} = 100\text{ cm} \times 100\text{ cm} = 10\,000\text{ cm}^{2}$ : multiply by $10\,000$ to change $\text{m}^{2}$ to $\text{cm}^{2}$ , divide by $10\,000$ to change back.
$1\text{ m}^{3} = 100\text{ cm} \times 100\text{ cm} \times 100\text{ cm} = 1\,000\,000\text{ cm}^{3}$ : multiply by $1\,000\,000$ to change $\text{m}^{3}$ to $\text{cm}^{3}$ , divide to change back.
$1\text{ cm}^{3} = 1\text{ mL}$ , $1000\text{ cm}^{3} = 1\text{ L}$ and $1\text{ m}^{3} = 1000\text{ L}$ .
A smaller unit gives a bigger number; check the direction of every conversion.
With lengths in different units, change them to one unit first, then multiply; choose the unit the answer asks for.
To compare or order quantities, change them all to one unit.
Count tiles or boxes along each edge, then multiply: round down for whole tiles that fit without cutting (and for boxes), round up for tiles cut to cover the whole floor. Dividing areas or volumes works only when they fit exactly.
Capacity is found from the inside measurements of a container.
Round a number of things to buy up (boxes of tiles, bags of soil), and a number of things that fit down.