Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 7 Unit 7, lesson 5 (volume of cuboids and prisms): a box is 6 cm by 4 cm by 5 cm. What is its volume?
Answer
120 cm³. The layers are 6 × 4 = 24 cm² and there are 5 of them: 24 × 5 = 120. - From this unit, lesson 1 (area of a trapezium): a trapezium has parallel sides of 3 cm and 7 cm and a height of 4 cm. What is its area?
Answer
20 cm². ½ × (3 + 7) × 4 = 5 × 4 = 20. - From this unit, lesson 5 (surface area of prisms): the end of a prism has an area of 10 cm² and a perimeter of 14 cm, and the prism is 5 cm long. What is its surface area?
Answer
90 cm². 2 × 10 + 14 × 5 = 20 + 70 = 90.
Words used in this lesson
- volume
- the amount of space inside a solid. For example: How much water a tank holds.
- cross-section
- the shape you see when you cut straight through a solid. For example: The end of a prism.
- cubic centimetre
- the volume of a cube 1 cm on each side. For example: A small sugar cube.
- litre
- 1000 cubic centimetres. For example: A bottle of water.
- capacity
- how much a container can hold. For example: The capacity of a tank is 60 litres.
- layer
- one slice of a prism, all the same shape. For example: One level of a stack of books.
How much water is in a channel?
A farmer in Rajasthan has dug an irrigation channel. Its cross-section is a trapezium: 2 m across the bottom, 4 m across the top and 1.5 m deep. A stretch of the channel is 100 m long. Predict: if the channel were 2 m wide all the way down it would hold 300 m³, and if it were 4 m wide all the way down, 600 m³. Where between those will the real channel be? Use the slider and the slices to see what the volume needs, and then come back to the channel.
The end is a triangle with area 20 cm² and the prism is 3 cm long. Each 1 cm slice has volume 20 cm³, and there are 3 of them: 20 × 3 = 60 cm³.
Show me the steps
- The end is a triangle with base 8 cm and height 5 cm, so its area is ½ × 8 × 5 = 20 cm². The prism is 3 cm long. The volume is 20 × 3 = 60 cm³.
- Press Show the slices. There are 3 slices, each 1 cm thick, and each has a volume of 20 × 1 = 20 cm³. Three of them: 60 cm³.
- Choose Trapezium and set the length to 5 cm. The end has area ½ × (6 + 10) × 4 = 32 cm², so the volume is 32 × 5 = 160 cm³.
- Choose L shape and set the length to 3 cm. The end is a rectangle 8 cm by 6 cm with a 4 cm by 3 cm piece cut out, 48 − 12 = 36 cm². The volume is 36 × 3 = 108 cm³.
The volume of a prism is the area of its cross-section times its length: V = A × L. It does not matter what the shape of the end is.
- Why: a slice 1 cm thick holds as many cubic centimetres as the end has square centimetres. A length of L cm has L slices.
- The units: a length is in cm, an area in cm² and a volume in cm³. A litre is 1000 cm³.
- The length is the distance between the two ends, measured along the sides at a right angle to them.
Back to the channel: the area of its cross-section is a trapezium, and the volume is worked out below.
Lengths, areas and volumes grow differently
Multiply every length of this prism by 2, then by 3, and watch the surface area and the volume. Are they multiplied by the same number as the lengths?
Every length is multiplied by 1. The longest side is 5 × 1 = 5 cm, the surface area is 48 × 1 = 48 cm² and the volume is 18 × 1 = 18 cm³.
Show me the steps
- The prism has a 3, 4, 5 end and is 3 cm long. Its surface area is 2 × 6 + 12 × 3 = 48 cm² and its volume is 6 × 3 = 18 cm³.
- Multiply every length by 2. The longest side is 10 cm. The surface area is 48 × 4 = 192 cm² and the volume is 18 × 8 = 144 cm³.
- Multiply every length by 3. The longest side is 15 cm. The surface area is 48 × 9 = 432 cm² and the volume is 18 × 27 = 486 cm³.
- Multiply every length by 4. The longest side is 20 cm. The surface area is 48 × 16 = 768 cm² and the volume is 18 × 64 = 1152 cm³.
Scaling: multiply every length by k, and the surface area is multiplied by k × k while the volume is multiplied by k × k × k.
A common mistake: multiplying every length you are given. A triangular prism has an end with a base of 6 cm, a height of 4 cm and two sloping sides of 5 cm, and a length of 10 cm. 6 × 4 × 5 × 10 = 1200 is not its volume. The area of the end uses only the base and the height: ½ × 6 × 4 = 12 cm². Then V = 12 × 10 = 120 cm³. Work out the area of the end on its own line first, and then multiply by the length once.
Worked examples
The channel: a trapezium cross-section 2 m across the bottom, 4 m across the top and 1.5 m deep, over a length of 100 m. Find the volume.
- Area of the cross-section: ½ × (2 + 4) × 1.5 = 3 × 1.5 = 4.5 m².
- Volume = 4.5 × 100 = 450 m³.
- That is between 300 and 600, at the middle width 3 m: 3 × 1.5 × 100 = 450.
A prism is 7 cm long. Its end is a right-angled triangle with the two shorter sides 9 cm and 4 cm. Find its volume.
- Area of the end: ½ × 9 × 4 = 18 cm².
- Volume = 18 × 7 = 126 cm³.
A fish tank in Manila is a box 40 cm by 30 cm by 50 cm. How many litres does it hold?
- The end is a rectangle: 40 × 30 = 1200 cm².
- Volume = 1200 × 50 = 60000 cm³.
- 1000 cm³ is 1 litre, so the tank holds 60000 ÷ 1000 = 60 litres.
Your go. The first step is given. A prism is 10 cm long. Its end is a trapezium with parallel sides of 5 cm and 9 cm and a height of 4 cm. Find its volume.
- Area of the end: ½ × (5 + 9) × 4 = 28 cm².
- Now multiply by the length, then check the units.
Check the full answer
Volume = 28 × 10 = 280 cm³. The units are cubic centimetres: an area multiplied by a length.
Check before you go on
1. A prism has an end with an area of 24 cm² and a length of 10 cm. What is its volume?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A triangular prism is 12 cm long. Its end is a triangle with a base of 6 cm and a height of 5 cm. What is its volume in cubic centimetres? The diagram is not drawn to scale.
3. A drinking trough on a farm in Mongolia is a prism 100 cm long. Its end is a trapezium with parallel sides of 30 cm and 50 cm and a depth of 20 cm. How many litres of water does it hold when full? One litre is 1000 cm³. The diagram is not drawn to scale.
Level 2 · Building
4. A prism has a volume of 450 cm³. Its end is a right-angled triangle with the two shorter sides 5 cm and 12 cm. How long is the prism, in centimetres?
5. A hexagonal candle in Hyderabad is a prism 10 cm tall. Its end is a regular hexagon with sides of 6 cm and an apothem of 5.2 cm. What is its volume in cubic centimetres?
6. A steel beam is a prism 15 cm long. Its end is an L shape: a rectangle 12 cm wide and 9 cm tall with a rectangle 5 cm wide and 4 cm tall cut out of one corner. What is its volume in cubic centimetres? The diagram is not drawn to scale.
Level 3 · Stretch
7. A swimming pool in Dubai is 10 m long and 5 m wide. It is 1 m deep at the shallow end and 3 m deep at the deep end, and the floor slopes evenly between them. How many cubic metres of water fill it? The diagram is not drawn to scale.
8. A toy box has a volume of 40 cm³. A larger box has the same shape, with every length 5 times as long. What is the volume of the larger box in cubic centimetres?
Something to think about
A box is 6 cm by 4 cm by 3 cm. Another box is 12 cm by 6 cm by 1 cm. Do they have the same volume? Do they need the same amount of card to make?
One way to think about it
Both hold 72 cm³, because 6 × 4 × 3 = 72 and 12 × 6 × 1 = 72. The first needs 2 × (24 + 18 + 12) = 108 cm² of card and the second needs 2 × (72 + 12 + 6) = 180 cm². Volume and surface area are different quantities, and the same volume can have very different surface areas. It is why a flat box and a cube of the same volume are packed differently.
Exit ticket
9. Kemal says: “A triangular prism has an end with an area of 15 cm² and a length of 4 cm. Its volume is ½ × 15 × 4 = 30 cm³, because the end is a triangle.” Which sentence is right?