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Surface area and volume
Cubes and cuboids, including counting hidden blocks.
Video coming soon
The written summary and downloads below are complete and ready to use.
A box, a tank and a building block all take up space. The volume of a shape is how much space it takes up. The surface area is the total area of all its outside faces, like the wrapping paper that would cover it. In this lesson you will learn how to:
- find a volume by counting cubes, including the ones you cannot see
- find the volume of a cuboid: length × width × height
- work backwards to a missing length
- find the surface area of a cube or a cuboid
- solve puzzles about painted cubes and boxes packed with blocks
Dark blue: whole numbers
Red dot: the decimal point
Blue: digits after the point
1. Volume: counting cubes
Key idea
We measure volume in cubes. A cube 1 cm long, 1 cm wide and 1 cm high has a volume of 1 cubic
centimetre, written 1 cm³. Bigger volumes, like a room, are measured in cubic metres
(m³). To find the volume of a shape made of cubes, count the cubes layer by layer, starting
at the bottom.
The hidden cubes rule
Cubes cannot float. Every cube sits on the table or on another cube. So if you can see a cube high up,
there must be cubes under it, even when you cannot see them. In our pictures, nothing else is
hiding at the back.
Example 1: how many cubes are in this shape?
- The bottom layer: the front left cube, and 3 more under the high cubes (one of them is hidden):4
- The top layer:3
- Add the layers:4 + 3 = 7
Answer7 cubes. If each cube is 1 cm³, the
volume is 7 cm³.
Example 2: how many cubes make this staircase?
- Every column stands on the table, so the bottom layer has a cube in all 6 places:6
- The middle layer: every column that is 2 or 3 cubes high:4
- The top layer: the columns that are 3 cubes high:2
- Add the layers:6 + 4 + 2 = 12
Answer12 cubes. You can only see 9 of them.
Watch out
Do not just count the cubes you can see. Count each layer, and remember the cubes underneath.
Look underneath
How to play
- Count the cubes and type your answer. Press Lift the layers to pull the layers apart and see the hidden cubes.
- Two with no slips moves you up a level. Level 1 has small shapes, level 2 has hidden cubes, and on level 3 you find how many more cubes would make the shape into a cuboid.
2. Volume of a cuboid
Key idea
A cuboid is a box shape: every face is a rectangle. It is made of layers of cubes. The bottom layer
is length × width cubes, and the height tells you how many layers there are. So
volume = length × width × height. A cube has every edge the same length, so its
volume is edge × edge × edge.
Example 1: what is the volume of this cuboid?
- One layer is 4 cubes long and 3 cubes wide:4 × 3 = 12
- There are 2 layers:12 × 2 = 24
AnswerThe volume is 24 cm³.
Example 2: what is the volume of this cuboid?
- Length × width:8 × 3 = 24
- × height:24 × 4 = 96
AnswerThe volume is 96 cm³.
Example 3: a cube has edges 5 cm long. What is its volume?
- Every edge is 5 cm:5 × 5 × 5 = 125
AnswerThe volume is 125 cm³. (125 is a cube
number: you met cube numbers in Lesson 1.7.)
Watch out
Volume multiplies all three lengths: 8 + 3 + 4 is not a volume, and nor is 8 × 3. Write the
unit with a small 3: cm³ or m³.
Cuboid builder
How to play
- Drag the red dots (or use the − and + buttons) to make the cuboid longer, wider or taller. Its volume changes as you go. Press Check when it matches the challenge.
- Two with no slips moves you up a level. Level 1 gives you a volume to make, level 2 also gives you one of the lengths, and level 3 asks for cubes, and for a missing height.
3. Working backwards: missing lengths
Key idea
If you know the volume and two of the lengths, find how many cubes are in one layer, then
divide: volume ÷ (length × width) = height. For a cube, ask: which number,
times itself and times itself again, makes the volume?
Example 1: this cuboid has a volume of 60 cm³. How high is it?
- One layer:5 × 3 = 15
- How many layers of 15 make 60?60 ÷ 15 = 4
AnswerIt is 4 cm high.
Example 2: 200 cm³ of water is poured into this tank. How deep is the water?
- One layer of water 1 cm deep:10 × 5 = 50
- How many layers of 50 make 200?200 ÷ 50 = 4
AnswerThe water is 4 cm deep.
Example 3: a cube has a volume of 64 cm³. How long is each edge?
- Try the cube numbers: 3 × 3 × 3 = 27 is too small.4 × 4 × 4 = 64
AnswerEach edge is 4 cm long.
Watch out
Divide by both lengths you know, not just one: 60 ÷ 5 = 12 is the area of the side, not the
height. And for a cube, do not divide by 3: 64 ÷ 3 is not the edge.
Fill the tank
How to play
- Follow the steps and type each number. When you have finished, the water pours in (or the stone drops in) so you can check.
- Two with no slips moves you up a level. Level 1 fills small tanks with cubes of water, level 2 has bigger tanks, and on level 3 a stone is dropped in and the water rises.
4. Surface area
Key idea
The surface area is the total area of all the faces on the outside. A cuboid has 6 faces in
3 matching pairs: the front and the back, the top and the bottom, and the two ends. You can only
see 3 of them, so find the area of each face you can see, and count each one twice. A cube has
6 faces that are all the same square: 6 × edge × edge. Surface area is an area, so it is
in cm² or m².
Example 1: find the surface area of this cuboid.
- The front, 5 by 2 (and the back is the same):5 × 2 = 10
- The top, 5 by 3 (and the bottom is the same):5 × 3 = 15
- The end, 3 by 2 (and the other end is the same):3 × 2 = 6
- Add all six faces:10 + 10 + 15 + 15 + 6 + 6 = 62
AnswerThe surface area is 62 cm².
Example 2: a cube has edges 4 cm long. What is its surface area?
- One face is a square, 4 by 4:4 × 4 = 16
- There are 6 faces:6 × 16 = 96
AnswerThe surface area is 96 cm².
Watch out
Count all 6 faces, not just the 3 you can see: 10 + 15 + 6 = 31 is only half of the box. And do
not mix the two up: volume fills the inside (cm³), surface area covers the outside
(cm²).
Wrap the box
How to play
- Tap a face of the box. Its hidden partner on the other side lights up with a dashed outline. Type the area of the face, and both get wrapped. When every face is wrapped, add them all up.
- Two with no slips moves you up a level. Level 1 has squares to count, level 2 has labelled boxes, and level 3 has boxes with no lid.
5. Hidden faces: the painted cube puzzle
Key idea
A big cube made of small cubes is dipped in paint. Only the outside gets painted. A small cube on a
corner has 3 painted faces. One along an edge (not a corner) has 2. One in the
middle of a face has 1. One hidden inside has none.
Example 1: this 3 by 3 by 3 cube was dipped in paint. Sort its 27 small cubes.
- A cube has 8 corners:8 with 3 painted faces
- 12 edges, each with 1 cube between its corners:12 with 2 painted faces
- 6 faces, each with 1 cube in the middle:6 with 1 painted face
- The one in the very middle, hidden inside:1 with none
- Check they make 27:8 + 12 + 6 + 1 = 27
Example 2: three 1 cm cubes are joined in a row. What is the surface area?
- Three cubes have 6 faces each:3 × 6 = 18
- There are 2 joins, and each join hides 2 faces (one on each cube):18 − 4 = 14
Answer14 faces are on the outside, so the surface
area is 14 cm². Check: 3 on the front, 3 on the back, 3 on top, 3 underneath and 1 at each end.
Watch out
Faces where cubes touch are hidden, so they are not part of the surface area. And remember the cubes in the
middle that you cannot see at all.
The paint pot
How to play
- Answer the question about the painted cube. Then tap the buttons to light up the small cubes with 3, 2, 1 or no painted faces. Lift the layers to see inside.
- Two with no slips moves you up a level. Level 1 asks about corners and how many cubes there are, level 2 about edges and faces, and level 3 about the cubes hidden inside.
6. How many fit in a box?
Key idea
To find how many blocks fit in a box, work out how many fit along the length, across the
width and up the height. Then multiply. If the blocks do not fit exactly, round each one
down: you cannot use part of a block, so there will be gaps.
Example 1: how many 2 cm cubes fit in this box?
- Along: 6 ÷ 2, across: 4 ÷ 2, up: 4 ÷ 2:3, 2 and 2
- Multiply:3 × 2 × 2 = 12
Answer12 cubes fit.
Example 2: how many 2 cm cubes fit in this box?
- Along: 7 ÷ 2 = 3 r 1, so only 3 fit:3
- Across: 4 ÷ 2:2
- Up: 5 ÷ 2 = 2 r 1, so only 2 fit:2
- Multiply:3 × 2 × 2 = 12
Answer12 cubes fit, with gaps left over.
Watch out
Do not just divide the volumes. The box in Example 2 holds 7 × 4 × 5 = 140 cm³ and each
cube is 8 cm³, but 140 ÷ 8 = 17.5 is wrong, because the gaps cannot be used.
Pack the box
How to play
- Work out how many blocks fit along, across and up, then how many altogether. Each row of blocks drops into the box as you go.
- Two with no slips moves you up a level. On level 1 the cubes fit exactly, on level 2 there are gaps, and level 3 packs bricks instead of cubes.
7. Volume and surface area problems
Key idea
First decide: is the question about the space inside (filling, water, sand, packing: volume) or about
covering the outside (paint, wrapping paper, card: surface area)? Then draw a quick sketch and
label it.
Example 1: a sandpit is 3 m long, 2 m wide and 1 m deep. Sand costs £20 for each cubic metre. How much does it cost to fill?
- Filling is about the inside, so find the volume:3 × 2 × 1 = 6
- 6 m³ at £20 each:6 × 20 = 120
Answer£120
Example 2: a tank is 20 cm long and 10 cm wide. A stone is dropped in and the water rises by 3 cm. What is the volume of the stone?
- The stone pushes up a layer of water 3 cm deep:20 × 10 × 3 = 600
AnswerThe stone has a volume of 600 cm³.
Watch out
Read the question again at the end. Did it ask for the volume, the surface area, a missing length, a
number of blocks or a cost?
Volume and surface area workshop
How it works
- Follow the steps and type each number. The . key is the decimal point.
- Two with no slips moves you up a level. Level 1 is volumes, level 2 works backwards, level 3 is surface areas, level 4 has sandpits, paint and tanks, and level 5 has puzzles that take two or three steps.
8. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Counting only the cubes you can see. count layer by layer: every cube high up has cubes under it.
- Adding the lengths for a volume. Saying a 8 cm by 3 cm by 4 cm cuboid has a volume of 15 cm³. multiply all three: 8 × 3 × 4 = 96 cm³.
- Multiplying only two lengths. length × width is one layer; multiply by the height for all the layers.
- The wrong unit. Writing a volume in cm², or a surface area in cm³. volume in cm³ or m³; surface area in cm² or m².
- Counting only 3 faces for a surface area. a cuboid has 6 faces: every face you can see has a partner round the back.
- Dividing by 3 to find a cube’s edge. find the number that times itself and times itself again makes the volume: 4 × 4 × 4 = 64.
- Dividing the volumes to see how many blocks fit. count how many fit along each edge, round each one down, then multiply.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 6.4.
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