Prism/cylinder volume and capacity | Year 9

01 / LEARN

Understand the idea.

Layers make volume

Volume counts the space inside a solid. A right prism has the same cross-section throughout its length: multiplying the cross-sectional area $B$ by the perpendicular length $L$ gives $V = B L$ . A cylinder uses circular cross-section area $π r^{2}$ .

Choose the distance perpendicular to the cross-section, rather than a sloping edge. Square units multiplied by length units produce cubic units. For liquid capacity, use the internal dimensions because the container walls do not hold liquid.

V = B L, V_{c y l i n d e r} = π r^{2} h

Volume and liquid capacity

Volume and capacity describe the same available space in different units. Each $1 {cm}^{3}$ holds $1 mL$ , and $1000 {cm}^{3}$ holds $1 L$ . A cubic metre holds $1000 L$ , not $100 L$ .

For a partly full upright tank, multiply the horizontal base area by the water depth. If volume is known instead, divide it by the base area to find depth. Convert the volume to match the base-area units first.

\begin{aligned} 1 {cm}^{3} & = 1 mL \\ 1000 {cm}^{3} & = 1 L \\ 1 m^{3} & = 1000 L \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A cylindrical container

Find the volume of a cylinder with internal radius 4 cm and height 10 cm. Give its capacity to the nearest millilitre.

Worked solution

1
Find the circular base area.
$B = π (4^{2}) = 16 π {cm}^{2}$
The internal radius is $4 cm$ .
2
Multiply by the height.
$V = 16 π \times 10 = 160 π {cm}^{3}$
The cylinder has the same circular cross-section through its height.
3
Convert volume to capacity.
$160 π {cm}^{3} = 160 π mL$
One cubic centimetre holds one millilitre.
4
Round to a whole millilitre.
$160 π \approx 502.6548 \Rightarrow 503 mL$
The tenths digit is $6$ , so the nearest whole millilitre is $503$ .

Answer

$160 π$ cm³; approximately 503 mL.

EXAMPLE 02

Find the water depth

A rectangular tank has an internal base 80 cm by 50 cm. It contains 40 L of water. Find the water depth.

Worked solution

1
Convert the water volume.
$40 L = 40 \times 1000 = 40000 {cm}^{3}$
The base dimensions are in centimetres, so cubic centimetres keep the units consistent.
2
Calculate the internal base area.
$B = 80 \times 50 = 4000 {cm}^{2}$
The horizontal base is rectangular.
3
Divide volume by base area.
$h = \frac{V}{B} = \frac{40000}{4000} = 10 cm$
Rearranging $V = B h$ gives the water depth.
4
Check the volume at that depth.
$4000 \times 10 = 40000 {cm}^{3} = 40 L$
The calculated depth holds exactly the stated water volume.

Answer

10 cm

03 / YOUR TURN

Try it yourself.

Check your understanding

What is the capacity in litres of 2500 cm³?

Show answer & reasoning

2.5 L.

Since $1000 {cm}^{3} = 1 L$ , divide by $1000$ : $2500 \div 1000 = 2.5 L$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Find the volume of a rectangular prism 5 cm by 3 cm by 8 cm.
2 marks
A triangular prism has a cross-section with base 6 cm and perpendicular height 4 cm. Its length is 9 cm. Find its volume.
2 marks
A cylinder has radius 3 cm and height 20 cm. Give its exact volume and its capacity to the nearest millilitre.
3 marks
A rectangular tank has internal dimensions 0.8 m by 0.5 m by 0.4 m. Find its capacity in litres.
2 marks
A small pool is 2 m long, 1.5 m wide and 0.6 m deep. How many litres fill it?
2 marks
A tank has a 50 cm by 40 cm internal base. Find the water depth when it contains 13 L.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the volume of a rectangular prism 5 cm by 3 cm by 8 cm.
2 marks
$V = 5 \times 3 \times 8 = 120$ cm³.
120 cm³
A triangular prism has a cross-section with base 6 cm and perpendicular height 4 cm. Its length is 9 cm. Find its volume.
2 marks
$B = \frac{1}{2} (6) (4) = 12$ cm²; $V = 12 (9) = 108$ cm³.
108 cm³
A cylinder has radius 3 cm and height 20 cm. Give its exact volume and its capacity to the nearest millilitre.
3 marks
$V = π (3^{2}) (20) = 180 π$ cm³ $\approx 565.49$ cm³.
This is about 565 mL.
$180 π$ cm³; 565 mL.
A rectangular tank has internal dimensions 0.8 m by 0.5 m by 0.4 m. Find its capacity in litres.
2 marks
$V = 0.8 \times 0.5 \times 0.4 = 0.16$ m³.
$0.16 \times 1000 = 160$ L.
160 L
A small pool is 2 m long, 1.5 m wide and 0.6 m deep. How many litres fill it?
2 marks
$V = 2 \times 1.5 \times 0.6 = 1.8$ m³ $= 1800$ L.
1800 L
A tank has a 50 cm by 40 cm internal base. Find the water depth when it contains 13 L.
3 marks
$13$ L $= 13000$ cm³. Base area $= 2000$ cm².
$h = \frac{13000}{2000} = 6.5$ cm.
6.5 cm

Checking saved progress…

Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Year 9 surface area and volume of right prisms and cylinders, capacity, absolute/relative/percentage measurement error, precision and practical measurement problems.

Official curriculum source ↗

In this lesson

Understand the idea.

Layers make volume

Volume and liquid capacity

See the method in action.

A cylindrical container

Find the circular base area.

Multiply by the height.

Convert volume to capacity.

Round to a whole millilitre.

Find the water depth

Convert the water volume.

Calculate the internal base area.

Divide volume by base area.

Check the volume at that depth.

Try it yourself.

Check your understanding

Keep practising