Prisms & cylinders · Ages 13–14
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 equal ends (its cross-section), 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 cuboid in the picture is 8 cm long, 12 cm wide and 7 cm high. What is its total surface area, in cm²?
Show the answer and working
Answer: 472 cm²
Three pairs of equal faces: 2 × (84 + 96 + 56) = 472 cm².
Example 2
The prism in the picture has a trapezium as its cross-section, 10 cm along the top and 22 cm along the bottom, each slanting side 10 cm; it is 15 cm long. What is the area of all its sides together, in cm²?
Show the answer and working
Answer: 780 cm²
The sides open into one rectangle: the perimeter 10 + 22 + 10 + 10 = 52 cm, times the length 15: 780 cm².
Example 3
A prism's cross-section has an area of 9 cm² and a perimeter of 11 cm, and the prism is 13 cm long. Its total surface area is 161 cm².
True or false?
Show the answer and working
Answer: True
Total surface area = 2 × cross-section + perimeter × length = 2 × 9 + 11 × 13 = 161 cm².
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The cuboid in the picture is 7 cm long, 9 cm wide and 8 cm high. What is its total surface area, in cm²?
Answer:
2.The prism in the picture has a right-angled triangle with sides 6 cm, 8 cm and 10 cm as its cross-section, and is 9 cm long. What is its total surface area, in cm²?
Answer:
3.The prism in the picture has a trapezium as its cross-section, 10 cm along the top and 22 cm along the bottom, each slanting side 10 cm; it is 8 cm long. What is the area of all its sides together, in cm²?
Answer:
4.The prism in the picture has an isosceles triangle as its cross-section: 18 cm across, 12 cm high, its two equal sides 15 cm each; the prism is 12 cm long. What is its total surface area, in cm²?
Answer:
5.The prism in the picture has an L-shaped cross-section, 9 cm wide and 14 cm high, both arms 4 cm thick; it is 10 cm long. What is its total surface area, in cm²?
Answer:
All the answers
- 1. 382 cm²
- 2. 264 cm²
- 3. 416 cm²
- 4. 792 cm²
- 5. 612 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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