Prisms · Ages 12–13
Surface area of a prism
2 × cross-section area + perimeter × length.
How to do it
Method
2 × cross-section area + perimeter × length
2 × 6 + 12 × 10 = 132 cm²
Another way
Face by face
6 + 6 + 30 + 40 + 50 = 132
In full
The surface area is all the faces' areas added. A prism has two identical ends (its cross-section, the same shape and size), and its other faces are rectangles, each as wide as the prism is long. Open the sides out and they join into one big rectangle: as long as the cross-section's perimeter, as wide as the length. So total surface area = 2 × cross-section area + cross-section perimeter × length. A right-angled triangle of sides 3 cm, 4 cm and 5 cm, 10 cm long: 2 × (3 × 4 ÷ 2) + (3 + 4 + 5) × 10 = 12 + 120 = 132 cm². Or face by face: two triangles, 6 + 6, and three rectangles, 3 × 10 + 4 × 10 + 5 × 10 = 30 + 40 + 50, 132 cm² in all. Careful: don't leave out the two ends; a box with no lid has just one.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The prism in the picture has a right-angled triangle with sides 3 cm, 4 cm and 5 cm as its cross-section, and is 8 cm long. What is its total surface area, in cm²?
Show the answer and working
Answer: 108 cm²
The two ends 2 × (3 × 4 ÷ 2) = 12, the sides = perimeter × length = 12 × 8 = 96; 108 cm² in all.
Example 2
Open out the three sides of the prism in the picture and they join into one rectangle, 12 cm wide. How long is the rectangle?
- A. 68 cm
- B. 56 cm
- C. 84 cm
- D. 31 cm
Choose the right answer
Show the answer and working
Answer: B. 56 cm
The three sides laid end to end make the cross-section's perimeter: 7 + 24 + 25 = 56 cm.
Example 3
The prism in the picture has a trapezium as its cross-section, 4 cm along the top and 10 cm along the bottom, each slanting side 5 cm; it is 11 cm long. What is the area of all its sides together, in cm²?
Show the answer and working
Answer: 264 cm²
The sides open into one rectangle: the perimeter 4 + 10 + 5 + 5 = 24 cm, times the length 11: 264 cm².
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Open out the three sides of the prism in the picture and they join into one rectangle, 8 cm wide. How long is the rectangle?
- A. 6 cm
- B. 20 cm
- C. 7 cm
- D. 12 cm
Choose the right answer
Answer:
2.The prism in the picture is 13 cm long, its cross-section a right-angled triangle of sides 24 cm, 7 cm and 25 cm. The two ends make cm², the three sides cm², the total surface area cm².
Fill in each blank
Answer:
3.The prism in the picture has a trapezium as its cross-section, 5 cm along the top and 11 cm along the bottom, each slanting side 5 cm; it is 12 cm long. What is the area of all its sides together, in cm²?
Answer:
4.A prism's cross-section has an area of 25 cm² and a perimeter of 11 cm, and the prism is 20 cm long. Its total surface area is 270 cm².
True or false?
Answer:
5.The prism in the picture has an isosceles triangle as its cross-section: 6 cm across, 4 cm high, its two equal sides 5 cm each; the prism is 20 cm long. What is its total surface area, in cm²?
Answer:
All the answers
- 1. D. 12 cm
- 2. 168, 728, 896
- 3. 312 cm²
- 4. True
- 5. 344 cm²
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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