Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Unit 7, lesson 3 (area of triangles and parallelograms): a triangle has base 12 cm and height 7 cm. What is its area?
Answer
42 cm². ½ × 12 × 7 = 42. - From Unit 7, lesson 4 (compound shapes): a 9 m by 5 m rectangle has a 2 m by 2 m square cut from it. What is the area left?
Answer
41 m². 45 − 4 = 41. - From Unit 1, lesson 2 (powers, squares and roots): what is 3 × 3 × 3?
Answer
27. Three threes multiplied together.
Words used in this lesson
- volume
- the amount of space inside a solid, counted in unit cubes. For example: A box holds 36 cm³.
- cubic centimetre (cm³)
- the volume of a cube with edges of 1 cm. For example: One sugar cube is about 1 cm³.
- cuboid
- a box shape with six rectangular faces. For example: A brick or a shoe box.
- prism
- a solid with the same end shape all along its length. For example: A tent, a Toblerone box, a block of cheese.
- cross-section
- the shape you see when you cut straight through a solid. For example: The triangle at the end of a tent.
- layer
- one level of unit cubes in a box. For example: The bottom layer of a crate.
How many boxes fit in the crate?
Mateus runs a bakery in Recife and sends cakes out in identical boxes, stacked in crates. A crate holds 7 boxes along its length, 4 across and 3 layers high. He does not count the boxes one at a time: he counts the boxes in one layer, and then the layers. The same idea gives the volume of any solid, which is the amount of space inside it.
Volume is counted in cubes. A cubic centimetre (cm³) is a cube with edges of 1 cm. Predict first: a box is 6 cm long, 3 cm wide and 2 cm high. How many cubes make one layer, and how many layers are there? Work out the volume first. Then set the length, width and height sliders to 6, 3 and 2, fill the box and check.
One layer is 5 by 4 = 20 cubes. With 1 of 2 layers filled there are 20 cubes. The whole box is 5 × 4 × 2 = 40 cm³.
Show me the steps
- The box is 5 cm long, 4 cm wide and 2 cm high. One layer is 5 × 4 = 20 cubes, so with 1 layer filled there are 20 cubes.
- Fill the second layer. 20 × 2 = 40 cubes. The box is full, so its volume is 40 cm³.
- Make the height 5 cm and fill all 5 layers. 20 × 5 = 100 cubes, so the volume is 100 cm³.
- Make the length 6 and the width 3 with the height 5. One layer is 6 × 3 = 18 cubes, and 5 layers hold 18 × 5 = 90 cm³.
Volume of a cuboid: length × width × height. The length times the width is the number of cubes in one layer, and the height is the number of layers. A height that is not whole still works: a box 4 cm by 2 cm by 6.5 cm is 6 whole layers of 8 cubes and half a layer, 8 × 6.5 = 52 cm³.
- Any prism: a prism has the same end shape all the way along its length. A layer 1 cm thick holds as many cubic centimetres as the end has square centimetres (some cubes are cut into pieces, as some squares were for the triangle). Volume of a prism = area of the end × length.
- Units: volume is in cubic units: cm³, m³. Three lengths multiplied, so a cube in the unit. (Area has two lengths, so a square in the unit.)
- Same area, same volume: if two prisms have the same end area and the same length, they have the same volume, whatever their end shapes.
- Reading the drawings: the length of a prism is drawn shorter than it is, as in any sketch of a solid. Read the numbers, not the picture.
- The slanted side is not the height: the area of a triangular end uses the height at a right angle to the base, never a sloping side.
Back to the crate: 7 × 4 = 28 boxes in a layer, and 28 × 3 = 84 boxes in the crate. Mateus used the layer times the layers.
The end can be any shape
Put three different ends on the same length. Which has the most room inside? Check the area of each end before you slide anything.
The end is a rectangle 4 × 3 = 12 cm². The prism is 5 cm long, so its volume is 12 × 5 = 60 cm³.
Show me the steps
- The end is a rectangle 4 cm by 3 cm, area 12 cm². At a length of 5 cm the volume is 12 × 5 = 60 cm³.
- Choose Triangle. The end is ½ × 4 × 6 = 12 cm², the same area, so the volume is again 12 × 5 = 60 cm³, a different shape with the same space inside.
- Choose L-shape. The end is 5 × 4 − 2 × 2 = 16 cm², so the volume is 16 × 5 = 80 cm³.
- Make the length 10 cm. The L-shape's volume is 16 × 10 = 160 cm³: twice the length, twice the volume.
A common mistake: “The triangular prism has a triangle with base 14 cm and height 3 cm, and is 7 cm long, so its volume is 14 × 3 × 7 = 294 cm³.” The end of this prism is a triangle, and the area of a triangle is half of base times height. The end has area ½ × 14 × 3 = 21 cm², and the volume is 21 × 7 = 147 cm³. First find the area of the end, then multiply by the length.
Worked examples
A water tank is 9 m long, 5 m wide and 3 m deep. What is its volume?
- One layer: 9 × 5 = 45 cubes, each 1 m³.
- There are 3 layers: 45 × 3 = 135.
- Volume: 135 m³.
A triangular prism has an end with base 10 cm and height 4 cm, and is 9 cm long. Find its volume.
- Area of the end: ½ × 10 × 4 = 20 cm².
- Volume = area of end × length = 20 × 9 = 180 cm³.
Your go. The first step is given. A prism has an L-shaped end: 6 cm by 5 cm with a 2 cm by 2 cm piece missing from a corner. It is 4 cm long. Find its volume.
- Area of the end: 6 × 5 = 30 cm², and the missing piece is 2 × 2 = 4 cm².
- Now find the area of the L, multiply by the length, then check.
Check the full answer
The end is 30 − 4 = 26 cm². The volume is 26 × 4 = 104 cm³.
Check before you go on
1. A cuboid is 5 cm long, 4 cm wide and 3 cm high. What is its volume?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A box is 10 cm long, 6 cm wide and 4 cm high. What is its volume in cubic centimetres?
3. A cube has edges of 6 cm. What is its volume in cubic centimetres?
Level 2 · Building
4. A tank is 5 m long and 4 m wide and holds 60 m³ when full. How deep is it, in metres?
5. Find the volume of this prism in cubic centimetres.
6. The end of this prism is an L-shape. Find its volume in cubic centimetres.
Level 3 · Stretch
7. A fish tank is 80 cm long and 40 cm wide, and the water in it is 25 cm deep. One litre is 1000 cm³. How many litres of water are in the tank?
8. Every edge of a cube is made 2 times as long. How many times as big is its volume?
Something to think about
A cube has edges of 3 cm. A second cube has edges twice as long. Another has edges three times as long. Without a calculator, work out how many of the small cubes would fit in each of the others, and say what the pattern is. Then say how many small cubes fill a cube with edges 10 times as long.
One way to think about it
Twice as long: 6 × 6 × 6 = 216 cm³, and 216 ÷ 27 = 8 small cubes. Three times as long: 9 × 9 × 9 = 729 cm³, which is 27 small cubes. The count is 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27: the edge factor multiplied by itself three times. With edges 10 times as long, 10 × 10 × 10 = 1000 small cubes. Lengths scale by k, areas by k × k and volumes by k × k × k.
Exit ticket
9. Ibrahim finds the volume of a triangular prism with a triangle of base 9 cm and height 4 cm, and a length of 7 cm, as 9 × 4 × 7 = 252 cm³. What is wrong?