Middle Years

# 9.10 Volume and capacity

Lesson

Conversions from volume to capacity is the mathematical equivalent of asking

"How much water can we pour into this container?"

### What is capacity?

Capacity

Capacity is a measure of how much a container can hold. We normally associate capacity with liquids, but it can also be used as a measure of gasses and solids.

When measuring the capacity of containers for everyday use, we will often use either millilitres (mL) or litres (L). For larger containers like bathtubs or swimming pools, we can use kilolitres (kL).

We can convert between these units using the conversion equations:

• $1$1 L$=$=$1000$1000 mL
• $1$1 kL$=$=$1000$1000 L

#### Worked Examples

##### Example 1

Convert $14500$14500 mL to L.

Think: To convert from millilitres to litres, we want to find how many litres there are in $14500$14500 millilitres. Since $1$1 L$=$=$1000$1000 mL, we can find how many litres there by finding how many sets of $1000$1000 fit into$14500$14500.

Do: We can find how many sets of $1000$1000 fit into $14500$14500 using division.

 $14500$14500 mL $=$= $\frac{14500}{1000}$145001000​ L Divide by the conversion factor $14500$14500 mL $=$= $14.5$14.5 L Perform the division
##### Example 2

Convert $3.24$3.24 kL to L.

Think: To convert from kilolitres to litres, we want to find how many litres there are in $3.24$3.24 kilolitres. Since $1$1 kL$=$=$1000$1000 L, we know that the number of litres will be $1000$1000 times more than the number of kilolitres.

Do: We can find how many litres there are by multiplying the number of kilolitres by $1000$1000.

 $3.24$3.24 kL $=$= $3.24\times1000$3.24×1000 L Multiply by the conversion factor $3.24$3.24 kL $=$= $3240$3240 L Perform the multiplication

Reflect: When converting from a smaller unit to a larger unit (as seen in the first example) we divide by the conversion factor. When converting from a larger unit to a smaller unit (as seen in the second example) we multiply by the conversion factor.

#### Practice question

##### Question 1

Convert $99000$99000 L to kL.

### Converting from volume to capacity

When converting from volume to capacity, there is only one conversion equation that we need to use:

$1$1 cm3$=$=$1$1 mL

This tells us that every one cubic centimetre has a capacity of one millilitre. It also tells us that any conversion equations involving millilitres will work the same way for cubic centimetres.

$1000$1000 cm3$=$=$1$1 L

Now we have a way to convert from volume to capacity, we can start finding the capacity of containers.

#### Worked Examples

##### Example 1

A jewellery box has dimensions $4$4 cm, $8$8 cm and $3$3 cm. What is its capacity in millilitres?

Think: We can find the volume in cubic centimetres by multiplying the dimensions together. We can then convert this volume directly into capacity.

Do: Multiplying the dimensions of the box together gives us:

 Volume $=$= $4\times8\times3$4×8×3 Multiply the dimensions together Volume $=$= $96$96 Perform the multiplication

So the box has a volume of $96$96 cm3.

Since one cubic centimetre equates to one millilitre, we can simply change our unit to get:

Capacity$=$=$96$96 mL

##### Example 2

A cylindrical bucket has a radius of $9$9 cm and a height of $20$20 cm. What is its exact capacity in litres?

Think: We can find the volume in cubic centimetres using the volume formula for a cylinder $V=\pi r^2h$V=πr2h. We can then convert this volume into capacity.

Do: Substituting the given dimensions into the volume formula gives us:

 Volume $=$= $\pi\times9^2\times20$π×92×20 Substitute in the given values Volume $=$= $\pi\times81\times20$π×81×20 Evaluate the square Volume $=$= $1620\pi$1620π Perform the multiplication

So the cylinder has a volume of $1620\pi$1620π cm3. Since there are $1000$1000 cubic centimetres in one litre, we can divide our volume by $1000$1000 to find the capacity in litres.

 Capacity $=$= $\frac{1620\pi}{1000}$1620π1000​ L Divide by the conversion factor Capacity $=$= $1.62\pi$1.62π L Perform the division

#### Practice question

##### Question 2

A cylinder has a diameter of $12$12 cm and height of $70$70 cm.

1. Find the volume of the cylinder in cubic centimetres.

2. What is the capacity of the cylinder in litres?

### The capacity of a cubic metre

As mentioned before, the only conversion we need in order to convert between volume and capacity is the equality $1$1 cm3$=$=$1$1 mL.

With this in mind, how many litres are there in one cubic metre?

To find how many litres there are in one cubic metres, there are two conversions we need to make. First, we want to convert our cubic metre into cubic centimetres. Once we have done this, we can then convert to litres, like so:

 Volume $=$= $1$1 m3 Volume $=$= $1$1 m$\times$×$1$1 m$\times$×$1$1 m One cubic metre is a cube of side length $1$1 m Volume $=$= $100$100 cm$\times$×$100$100 cm$\times$×$100$100 cm Convert each dimension from metres to centimetres Volume $=$= $100\times100\times100$100×100×100 cm3 Combine the units to get cubic centimetres Volume $=$= $1000000$1000000 cm3 Perform the multiplication

These calculations tell us that one cubic metre is equal to $1000000$1000000 cubic centimetres. Now that our volume is in cubic centimetres, we can convert to capacity. We can convert using the equation $1$1 L$=$=$1000$1000 cm3:

 Capacity $=$= $\frac{1000000}{1000}$10000001000​ L Divide by the conversion factor Capacity $=$= $1000$1000 L Perform the division

So the capacity of one cubic metre is $1000$1000 litres.

To convert from capacity back into volume, we can simply reverse these steps.

#### Worked Example

A fish tank has a capacity of $6.58$6.58 kL. What is its volume in cubic metres?

Think: To find the number of cubic metres, we want to convert the capacity into litres and then convert from litres into cubic metres.

Do: When converting from kilolitres to litres the unit is getting smaller so we multiply by the conversion factor.

 $6.58$6.58 kL $=$= $6.58\times1000$6.58×1000 L Multiply by the conversion factor $6.58$6.58 kL $=$= $6580$6580 L Perform the multiplication

We can then convert from litres to cubic metres, remembering that one cubic metre equates to $1000$1000 litres. We can find how many cubic metres are filled by $6580$6580 litres by dividing by $1000$1000.

 $6580$6580 L $=$= $\frac{6580}{1000}$65801000​ m3 Division by $1000$1000 $6580$6580 L $=$= $6.58$6.58 m3 Perform the division

As such, the volume of the fish tank is $6.58$6.58 m3.

Reflect: Notice that the volume in cubic metres is equal to the capacity in kilolitres. This tells that one cubic metre equates to one kilolitre.

#### Practice question

##### Question 3

Kathleen is constructing a swimming pool designed to hold $34.4$34.4 kilolitres of water.

She has already decided on a base area of $8$8 square metres.

1. What will the volume of Kathleen's pool be?