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 6 (nets and surface area): a box is 5 cm by 4 cm by 3 cm. What is the area of its largest face?
Answer
20 cm². The faces are 5 × 4 = 20, 5 × 3 = 15 and 4 × 3 = 12. - From this unit, lesson 1 (area of a trapezium): a trapezium has parallel sides of 4 cm and 8 cm and a height of 3 cm. What is its area?
Answer
18 cm². ½ × (4 + 8) × 3 = 6 × 3 = 18. - From Year 7 Unit 7, lesson 3 (triangles and parallelograms): a triangle has a base of 12 cm and a height of 9 cm. What is its area?
Answer
54 cm². Half of 12 × 9 = 108 is 54.
Words used in this lesson
- prism
- a solid with two identical ends joined by flat rectangular sides. For example: A tent, a ramp.
- net
- a flat pattern that folds up into a solid. For example: A box before it is folded.
- face
- one flat surface of a solid. For example: The floor of a tent.
- surface area
- the total area of all the faces of a solid. For example: The fabric needed for a tent.
- cross-section
- the shape you see when you cut straight through a solid. For example: The triangle at each end of a tent.
- perimeter
- the distance all the way round a shape. For example: The edge of the tent's end.
How many pieces of fabric make a tent?
A hiking group in the Atlas Mountains has a tent shaped like a triangular prism: two triangular ends, and a floor and two sloping sides that are rectangles. Each end has a base of 6 m and a height of 4 m, the sloping sides of each end are 5 m, and the tent is 5 m long. Predict: how many separate pieces of fabric are there, and how many different shapes? Then use the first figure to build a net, and add the faces one at a time.
The net is resized to fill the frame, so compare the numbers, not the sizes.
The end is a right-angled triangle with sides 3 cm, 4 cm and 5 cm, and the prism is 5 cm long. The rectangles are 3 × 5 = 15, 4 × 5 = 20 and 5 × 5 = 25 cm², and each triangle is ½ × 3 × 4 = 6 cm². The surface area is 15 + 20 + 25 + 6 + 6 = 72 cm².
Show me the steps
- The end is a right-angled triangle with sides 3, 4 and 5 cm, and the prism is 5 cm long. The net has three rectangles, 3 × 5, 4 × 5 and 5 × 5, and two triangles.
- Press Add the next face five times. The areas are 15, 20, 25, then two triangles of ½ × 3 × 4 = 6 each. The total is 15 + 20 + 25 + 6 + 6 = 72 cm².
- Choose the triangle 5, 12, 13 and set the length to 4 cm. The rectangles are 20, 48 and 52 cm², which make 120 cm². The two triangles are 30 cm² each, 60 cm². The total is 180 cm².
- Set the length to 8 cm with the same triangle. The rectangles are 40, 96 and 104 cm², which make 240 cm², and the triangles are still 60 cm². The total is 300 cm². Only the rectangles grew.
The surface area of a prism is 2 × (area of the end) + (perimeter of the end) × length.
- Why: the two ends are identical, so they have the same area twice. The side faces are rectangles, each as long as the prism and as wide as one edge of the end, so together they unwrap into one rectangle that is as wide as the perimeter and as tall as the length.
- A net is the prism unfolded flat. Every face appears once, and the surface area is the total area of the net.
- The units are square units. The volume, found in the next lesson, is cubic units.
- Right prisms: every prism in this unit has its sides at right angles to its ends.
Back to the tent: its end is a triangle with base 6 m and height 4 m, and the fabric needed is worked out below.
Unwrap the sides
Choose each prism in turn and look at the long rectangle the sides make. How wide is it, and how tall? What happens to the surface area when the length of the prism changes and the end does not?
The end is a triangle with sides 3, 4 and 5 cm. Its area is 6 cm² and its perimeter is 12 cm. Two ends: 2 × 6 = 12 cm². The sides unwrap into a rectangle 12 cm by 6 cm: 72 cm². Total 84 cm².
Show me the steps
- The end is a right-angled triangle with sides 3, 4, 5 cm. Its area is 6 cm² and its perimeter 3 + 4 + 5 = 12 cm. The prism is 6 cm long. Two ends: 12 cm². Sides: 12 × 6 = 72 cm². Total 84 cm².
- Choose Trapezium 12, 6, with a length of 6 cm. The end has area ½ × (12 + 6) × 4 = 36 cm² and perimeter 12 + 5 + 6 + 5 = 28 cm. Two ends: 72 cm². Sides: 28 × 6 = 168 cm². Total 240 cm².
- Set the length to 3 cm with the same trapezium. The ends are still 72 cm². The sides are 28 × 3 = 84 cm². Total 156 cm². Halving the length did not halve the surface area.
- Choose Trapezium 11, 5 and set the length to 4 cm. The end has area ½ × (11 + 5) × 8 = 64 cm² and perimeter 11 + 10 + 5 + 8 = 34 cm. Two ends: 128 cm². Sides: 34 × 4 = 136 cm². Total 264 cm².
A common mistake: counting the sides but not the ends. For a prism with a 3, 4, 5 end and a length of 5 cm, the sides are 12 × 5 = 60 cm². That is not the surface area. Each end is ½ × 3 × 4 = 6 cm², so with both ends it is 60 + 6 + 6 = 72 cm². Counting one end gives 66, and that is wrong too. Before you add, count the faces on the net: a triangular prism has five.
Worked examples
A triangular prism is 10 cm long. Its end is a right-angled triangle with sides of 9, 12 and 15 cm. Find its surface area.
- Three rectangles: 9 × 10 = 90, 12 × 10 = 120 and 15 × 10 = 150, which is 360 cm².
- Two triangles: ½ × 9 × 12 = 54 each, which is 108 cm².
- Total: 360 + 108 = 468 cm².
A prism is 10 cm long. Its end is a trapezium with parallel sides of 6 cm and 2 cm, a height of 3 cm and a sloping side of 5 cm. Find its surface area.
- Area of one end: ½ × (6 + 2) × 3 = 12 cm², so two ends are 24 cm².
- Perimeter of the end: 6 + 5 + 2 + 3 = 16 cm.
- The sides: 16 × 10 = 160 cm². Total: 24 + 160 = 184 cm².
The tent: the ends have a base of 6 m and a height of 4 m, the sloping sides are 5 m, and the tent is 5 m long. How much fabric is in the whole tent, floor included?
- Two ends: 2 × ½ × 6 × 4 = 24 m².
- Perimeter of the end: 6 + 5 + 5 = 16 m, and the sides are 16 × 5 = 80 m² (floor and both slopes).
- Total: 24 + 80 = 104 m², from five pieces and three different shapes.
Your go. The first step is given. A triangular prism is 6 cm long. Its end is a right-angled triangle with sides of 8, 15 and 17 cm. Find its surface area.
- Area of one end: ½ × 8 × 15 = 60 cm².
- Now find the two ends together, the perimeter times the length, and add. Then check against a net.
Check the full answer
Two ends: 120 cm². Perimeter 8 + 15 + 17 = 40 cm, and 40 × 6 = 240 cm². Total 120 + 240 = 360 cm². Check with the net: rectangles 48 + 90 + 102 = 240, and two triangles 60 each.
Check before you go on
1. A prism has an end of area A and perimeter P, and a length L. Which gives its surface area?
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 right-angled triangle with sides of 6 cm, 8 cm and 10 cm. What is its surface area in square centimetres? The diagram is not drawn to scale.
3. A triangular prism is 9 cm long. Its end is an isosceles triangle with a base of 10 cm, a height of 12 cm and sloping sides of 13 cm. What is its surface area in square centimetres? The diagram is not drawn to scale.
Level 2 · Building
4. A prism is 15 cm long. Its end is a trapezium with parallel sides of 4 cm and 10 cm, a height of 4 cm and sloping sides of 5 cm each. What is its surface area in square centimetres? The diagram is not drawn to scale.
5. A triangular prism has a right-angled triangle for each end, with sides of 3 cm, 4 cm and 5 cm. Its surface area is 192 cm². How long is the prism, in centimetres?
6. A small tent in the Atlas Mountains has two triangular ends and two sloping sides, but no floor. The ends are isosceles triangles with a base of 4 m, a height of 1.5 m and sloping sides of 2.5 m, and the tent is 3 m long. How many square metres of fabric does it have? The diagram is not drawn to scale.
Level 3 · Stretch
7. A triangular prism has a right-angled triangle for each end, with sides of 3 cm, 4 cm and 5 cm. One prism is 10 cm long and another is 20 cm long. How many square centimetres more surface area does the longer prism have?
8. A gift box in Seoul is a prism 10 cm long. Its end is a regular hexagon with sides of 4 cm and an apothem of 3.5 cm. What is its surface area in square centimetres? The diagram is not drawn to scale.
Something to think about
A triangular prism has a 5, 12, 13 end and a length of 10 cm. Another prism has the same end and a length of 20 cm. Does the longer prism have double the surface area? Would it be closer to double if the lengths were 100 cm and 200 cm?
One way to think about it
No: 60 + 300 = 360 cm² for the first and 60 + 600 = 660 cm² for the second. Only the sides double. The two ends stay the same, 60 cm² in both. With lengths of 100 cm and 200 cm the surface areas are 3060 and 6060 cm², much closer to double, because the ends matter less and less.
Exit ticket
9. Dina says: “The surface area of this prism is ½ × 8 × 3 × 10 = 120 cm².” Which is the surface area of the prism? The diagram is not drawn to scale.