# Volume of Prisms and Capacity

> Find the volume of any prism and a cylinder, change between cubic centimetres, millilitres and litres, and work out how far water rises in a tank.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/volume-and-capacity/. Updated 4 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The fish tank
Noah gets a fish tank for his birthday. It is 40 cm long, 25 cm wide and 30 cm tall. "How many litres of water do I need?" he asks.

His aunt shows him. The bottom of the tank is $40 \times 25 = 1000$ square centimetres. Every centimetre of water on top of that bottom is 1000 cubic centimetres, which is 1 litre. So 30 cm of water is 30 litres.

Then Noah drops in a rock, and the water rises 1 cm. "Your rock takes up 1000 cubic centimetres," says his aunt.

**In short:** The volume of a prism is the area of its end times its length, so a box is length × width × height. One cubic centimetre holds 1 mL, so 1000 cm³ is 1 litre. When water rises in a tank, the extra volume is the base area times the rise.

## Before you start

- Area of rectangles and triangles, and volume of a box (see Perimeter, Area and Volume).
- Changing mL to L (see Metric Units and Conversions).

## Volume of a prism

A prism has the same shape all the way along (see 3D Shapes, Nets and Views). Its volume is the area of that shape times the length.

### Key idea: The prism rule
$$\text{volume} = \text{area of the end} \times \text{length}$$

For a rectangular prism (a box), the end is a rectangle, so volume $= l \times w \times h$.

### Worked example: A triangular prism
A tent is a triangular prism. Its triangle end has a base of 6 m and a height of 4 m, and the tent is 10 m long.

Area of the end $= \frac{1}{2} \times 6 \times 4 = 12$ m². Volume $= 12 \times 10 = 120$ m³.

### Worked example: An L shaped prism
A step has an L shaped end with an area of 30 cm², and it is 8 cm long. Volume $= 30 \times 8 = 240$ cm³. Split an L shape into two rectangles to find its area.

## A cylinder

A cylinder is like a prism with a circle for its end. The area of a circle is $\pi \times r \times r$, where $r$ is the radius and $\pi$ is about 3.14.

### Worked example: A water tank
A cylinder has a radius of 10 cm and a height of 20 cm.

Area of the circle $= 3.14 \times 10 \times 10 = 314$ cm². Volume $= 314 \times 20 = 6280$ cm³, which holds 6.28 L.

## Volume and capacity

**Capacity** is how much a container holds. The units fit together:

### Key idea: Volume to capacity
- 1 cm³ holds 1 mL.
- 1000 cm³ holds 1 L.
- 1 m³ holds 1000 L.

### Worked example: Litres in a tank
A tank is 30 cm by 20 cm by 10 cm. Its volume is $6000$ cm³, so it holds 6000 mL, which is 6 L.

A box 2 m by 1 m by 0.5 m has a volume of 1 m³. It holds 1000 L.

## Water that rises

### Key idea: Rise in water level
$$\text{volume added} = \text{base area} \times \text{rise}$$

So the rise is the volume added divided by the base area.

### Worked example: Pouring in
A tank has a base 40 cm by 25 cm, which is 1000 cm². Pour in 3 L, which is 3000 cm³. The water rises $3000 \div 1000 = 3$ cm.

### Worked example: A stone goes in
A tank has a base 20 cm by 15 cm. A stone is dropped in and the water rises 2 cm. The stone's volume is $20 \times 15 \times 2 = 600$ cm³.

### Challenge: Selective level: scaling and filling
Double every edge of a cube. The old cube is $1 \times 1 \times 1 = 1$. The new one is $2 \times 2 \times 2 = 8$. The volume is **8 times** as big, not 2 times.

A tank 50 cm by 40 cm by 30 cm is half full. Its full volume is $60\,000$ cm³, which is 60 L. It needs another 30 L to fill.

How many 2 cm cubes fit in a box 10 cm by 6 cm by 4 cm? Along each edge: $5$, $3$ and $2$ cubes. That is $5 \times 3 \times 2 = 30$ cubes.

**Mind map**

```text
Volume and capacity
  Prisms
    Area of end x length
    Box: l x w x h
    Triangle end: half b x h
  Cylinder
    Circle area: 3.14 x r x r
    Times the height
  Capacity
    1 cm³ = 1 mL
    1000 cm³ = 1 L
    1 m³ = 1000 L
  Water level
    Volume = base x rise
    Stone volume from rise
```

## Quick summary

- Volume of a prism $=$ area of the end $\times$ length.
- A box: $l \times w \times h$. A cylinder: $3.14 \times r \times r \times h$.
- 1 cm³ holds 1 mL, 1000 cm³ holds 1 L, and 1 m³ holds 1000 L.
- Water rise $=$ volume added $\div$ base area.
- An object's volume equals the volume of water it pushes up.
- Doubling every edge makes the volume 8 times bigger.

## Quiz

Try these questions from the chapter quiz.

1. What is the volume of a box 5 cm by 4 cm by 3 cm?
   - A. 12 cm³
   - B. 20 cm³
   - C. 60 cm³
   - D. 47 cm³
   - Answer: C. 60 cm³. $5 \times 4 \times 3 = 60$ cm³.

2. How many mL does a container of 250 cm³ hold?
   - A. 25 mL
   - B. 250 mL
   - C. 2500 mL
   - D. 0.25 mL
   - Answer: B. 250 mL. 1 cm³ holds 1 mL.

3. How many cubic centimetres hold 1 litre?
   - A. 10
   - B. 100
   - C. 1000
   - D. 10 000
   - Answer: C. 1000. 1000 cm³ holds 1000 mL, which is 1 L.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/volume-and-capacity/#quiz