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 5 (volume of cuboids and prisms): a box is 4 cm by 5 cm by 6 cm. What is its volume?
Answer
120 cm³. 4 × 5 = 20 for one layer, 20 × 6 = 120. - From Unit 7, lesson 3 (area of triangles and parallelograms): a triangle has base 9 cm and height 10 cm. What is its area?
Answer
45 cm². ½ × 9 × 10 = 45. - From Unit 1, lesson 2 (powers, squares and roots): what is 7²?
Answer
49. 7 × 7 = 49.
Words used in this lesson
- net
- a flat pattern that folds up into a solid. For example: A flattened cereal box.
- face
- one flat side of a solid. For example: A cube has six faces.
- surface area
- the total area of all the faces of a solid. For example: The card needed to make a box.
- edge
- where two faces meet. For example: A cube has twelve edges.
- cuboid
- a box with six rectangular faces. For example: A shoe box.
- open box
- a box with one face missing. For example: A tray or a pencil case without a lid.
How much card?
A tea company in Kericho packs its tea in closed card boxes. Before the first box is made, someone has to say how much card each one uses, and how the flat sheet should be cut. That flat shape is called a net: the box cut along some edges and opened out so every face lies in one plane. The total area of the faces is the surface area.
Predict first: a closed box has how many faces, and how many different sizes of face does a cuboid have? Then unfold it.
The box is 4 cm by 3 cm by 2 cm. Its top and base are 12 cm² each, its front and back 8 cm² each and its ends 6 cm² each. 2 × 12 + 2 × 8 + 2 × 6 = 52 cm².
Show me the steps
- The box is 4 cm by 3 cm by 2 cm. The top and base are each 4 × 3 = 12 cm², the front and back each 4 × 2 = 8 cm², and the two ends each 3 × 2 = 6 cm².
- Add the six faces: 12 + 12 + 8 + 8 + 6 + 6 = 52 cm². The same as 2 × (12 + 8 + 6) = 52 cm².
- Make the height 5 cm. The top and base stay 12 cm², the front and back become 4 × 5 = 20 cm², the ends 3 × 5 = 15 cm². The surface area is 2 × (12 + 20 + 15) = 94 cm².
- Make all three 4 cm, a cube. Every face is 4 × 4 = 16 cm², and six faces give 96 cm².
The surface area of a solid is the total area of all its faces, and a net shows every face.
- Cuboid: six faces in three equal pairs, so the surface area is 2 × (lw + lh + wh).
- Triangular prism: two triangular ends and three rectangles. The rectangles join in one strip as long as the prism, whose width is the triangle's perimeter. Surface area = 2 × area of the end + perimeter of the end × length. The rectangles use the sides of the triangle, sloping ones included; the triangles use the perpendicular height.
- One solid, many nets: a solid can be unfolded in different ways, but every net of the same solid has the same faces, so the same surface area.
- Units: surface area is an area, so cm² or m², never cm³.
- Reading the drawings: the length of a solid is drawn shorter than it is, as in any sketch. Read the numbers, not the picture.
Back to the tea boxes: the net's total area is the card needed, and the volume (lesson 5) is how much tea fits. They are different numbers: the same volume can need more or less card.
A prism with a triangle at each end
Now a triangular prism. Its net has three rectangles in a row, each as wide as one side of the triangle, and two triangles. Notice what the three rectangles make when they are side by side.
The ends are triangles with sides 3, 4 and 5 cm. Each triangle is ½ × 3 × 4 = 6 cm², so two are 12 cm². The three rectangles are (3 + 4 + 5) × 6 = 72 cm². The surface area is 12 + 72 = 84 cm².
Show me the steps
- Triangle with sides 3, 4 and 5 cm, prism 6 cm long. Each end is ½ × 3 × 4 = 6 cm², so the two ends are 12 cm². The rectangles are 3 × 6, 4 × 6 and 5 × 6, which together are (3 + 4 + 5) × 6 = 72 cm². Total 84 cm².
- Make the length 10 cm. The ends stay 12 cm², and the rectangles are 12 × 10 = 120 cm². The total is 132 cm².
- Choose sides 6, 8, 10 with length 10 cm. Each end is ½ × 6 × 8 = 24 cm², so two ends are 48 cm². The rectangles are (6 + 8 + 10) × 10 = 240 cm². Total 288 cm².
- Choose sides 5, 12, 13 with length 4 cm. Each end is ½ × 5 × 12 = 30 cm², so two ends are 60 cm². The rectangles are (5 + 12 + 13) × 4 = 120 cm². Total 180 cm².
- Choose sides 5, 5, 6 with length 10 cm. This end has no right angle: the height to the 6 cm side is 4 cm, so each end is ½ × 6 × 4 = 12 cm² and the two ends are 24 cm². The rectangles use all three sides, the sloping ones too: (5 + 6 + 5) × 10 = 160 cm². Total 184 cm².
A common mistake: “I can see three faces of the box, 6 by 3, 6 by 2 and 3 by 2, so the surface is 18 + 12 + 6 = 36 cm².” A closed box has six faces. The three you cannot see are the same as the three you can: the base is like the top, the back like the front. The surface area is 36 × 2 = 72 cm². A net lays out every face so none are missed.
Which patterns fold into a cube?
A net must have the right squares in the right places. Six squares joined edge to edge are not always a net. Two quick tests say a pattern does not fold: a row of five or six never folds, because the fifth square would land on the first, and a block of four squares in a 2 by 2 square never folds either.
A test that says a pattern does fold uses opposite faces. In a straight row of squares, squares two places apart are opposite faces of the cube. A cube has three pairs of opposite faces, and every square must be in exactly one pair. Try pattern P below: in its row of four, squares 1 and 3 are opposite and squares 2 and 4 are opposite. The two single squares are joined on opposite sides of the row, so they are the top and the base. That makes three pairs, so P folds. Now try Q: it has no row of five and no 2 by 2 block, but both extra squares are joined below the row, so both fold onto the base and the top is missing. Q does not fold.
Worked examples
A closed box is 8 cm long, 5 cm wide and 2 cm high. Find its surface area.
- Top and base: 8 × 5 = 40 cm² each.
- Front and back: 8 × 2 = 16 cm² each. Ends: 5 × 2 = 10 cm² each.
- One of each: 40 + 16 + 10 = 66. Two of each: 2 × 66 = 132 cm².
A wedge is a triangular prism 9 cm long. Its ends are right-angled triangles with sides 3 cm, 4 cm and 5 cm. Find its surface area.
- Two ends: each is ½ × 3 × 4 = 6 cm², so 2 × 6 = 12 cm².
- Three rectangles, one for each side of the triangle, each 9 cm long: (3 + 4 + 5) × 9 = 12 × 9 = 108 cm².
- Total: 12 + 108 = 120 cm².
Your go. The first step is given. A wedge-shaped door stop is a triangular prism. Its ends are right-angled triangles with sides 12 cm, 16 cm and 20 cm, and it is 5 cm long. Find its surface area.
- Area of one end: ½ × 12 × 16 = 96 cm².
- Now find the two ends together, then the rectangles, then the total, then check.
Check the full answer
Two ends: 2 × 96 = 192 cm². Rectangles: (12 + 16 + 20) × 5 = 48 × 5 = 240 cm². Total 432 cm².
Check before you go on
1. A closed box is 5 cm long, 3 cm wide and 2 cm high. Which list shows the areas of all six faces, in square centimetres?
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 cube has edges of 7 cm. What is its surface area in square centimetres?
3. A closed box is 10 cm long, 6 cm wide and 5 cm high. What is its surface area in square centimetres?
Level 2 · Building
4. The ends of this triangular prism are isosceles triangles. Find its surface area in square centimetres.
5. A tray is an open-topped box with no lid. Its base is 12 cm by 8 cm and it is 3 cm deep. How many square centimetres of card make the tray?
6. Which of these patterns of six squares folds up to make a cube?
Level 3 · Stretch
7. A closed box has a square base 6 cm by 6 cm and a total surface area of 240 cm². How tall is the box, in centimetres?
8. Two closed boxes each hold 36 cm³. One is 4 cm by 3 cm by 3 cm and the other is 9 cm by 4 cm by 1 cm. How many more square centimetres of card does the flatter box need?
Something to think about
A closed box 1 cm by 1 cm by 24 cm and a closed box 2 cm by 3 cm by 4 cm each hold 24 cm³ of rice. Which needs more card, and by how much? Why might a company choose the second shape, and why might it still choose the first?
One way to think about it
The long thin box has surface area 2 × (1 + 24 + 24) = 98 cm². The 2 by 3 by 4 box has 2 × (6 + 8 + 12) = 52 cm². The thin box needs 46 cm² more card. The compact box wastes less card and is cheaper to make and ship. A company might still choose the thin shape if it has to fit a narrow shelf or a letterbox, so the cheapest card is not always the best decision.
Exit ticket
9. Ayo says the surface area of a closed box 6 cm by 4 cm by 3 cm is 6 × 4 × 3 = 72 cm². What is wrong?