A prism has the same cross-section all the way along its length. To find the total surface area, unfold it into a net, find each face’s area and add them. The faces are two identical ends and a set of rectangles around the sides.
This lesson belongs to area, perimeter and surface area. It uses rectangles and triangles from decomposing compound shapes and unit care from converting units.
How do you find the area of the side faces quickly?
The rectangles wrapped around a prism join into one strip when you unfold it. The strip is as long as the perimeter of the cross-section and as wide as the length of the prism.
So: surface area = 2 × (area of one end) + (perimeter of the end) × (length). This shortcut works for any prism, including a triangular one, an L-shaped one or a trapezium prism.
Step by step
- Sketch the net and label each face.
- Find the area of one end face and double it.
- Find the perimeter of the end face. Include the sloping side.
- Multiply that perimeter by the length of the prism.
- Add the two results.
- Write the unit as cm² or m².
Worked example
A triangular prism has a right-angled triangle as its cross-section with sides 6 cm, 8 cm and 10 cm (10 cm is the longest side). The prism is 15 cm long. Find the total surface area.
Ends. Area of one triangle = ½ × 6 × 8 = 24 cm². Two ends = 48 cm².
Rectangles. Their widths are 6, 8 and 10, all with length 15. Perimeter of the triangle = 6 + 8 + 10 = 24 cm. Strip area = 24 × 15 = 360 cm².
Total. 48 + 360 = 408 cm².
Check face by face. 6 × 15 = 90, 8 × 15 = 120, 10 × 15 = 150. 90 + 120 + 150 = 360, and 360 + 48 = 408. The two methods agree.
The mistake to watch for
A student finds one triangle end and forgets the other.
Mistaken answer: 24 + 360 = 384 cm²
A prism has two ends. Only one was counted.
A second common slip is to use the slant side 10 cm as the triangle’s height. The height in ½ × base × height must be at right angles to the base, which here is the side 8 cm. Drawing the net and ticking off each face guards against both errors.
Check yourself
1. Find the total surface area of a cuboid 5 cm by 4 cm by 3 cm.
Show answer
2 × (5 × 4 + 5 × 3 + 4 × 3) = 2 × (20 + 15 + 12) = 2 × 47 = 94 cm².
2. A closed cylinder has radius 3 cm and height 10 cm. Find its total surface area.
Show answer
Two circles: 2 × π × 3² = 18π. Curved surface: 2π × 3 × 10 = 60π. Total = 78π = 245.04, so 245 cm² (3 s.f.).
3. An open box (no lid) is 10 cm long, 6 cm wide and 4 cm high. Find the area of card needed.
Show answer
Base: 10 × 6 = 60. Two long sides: 2 × 10 × 4 = 80. Two short sides: 2 × 6 × 4 = 48. Total = 60 + 80 + 48 = 188 cm².
Where this leads next
Finish the module with checking an area answer using dimensions, then attempt the mixed practice set. The non-calculator working trainer helps you practise the multi-step arithmetic.
A student who can do the arithmetic but misses a face benefits from someone checking the sketch. That kind of feedback is what we offer in online one-to-one Mathematics tuition.