What is the total surface area of a cuboid? You may refer to the net of the cuboid given in the figure.
Answer: Total surface area (TSA) of a cuboid = 2(wl + hl + hw), the total area of its six rectangular faces.
Step-by-step solution
Idea: Unfold the cuboid into its net. The net is six rectangles, and they come in three equal pairs, so add the three different areas and double.
- A cuboid has 6 faces, all rectangles. Opposite faces are the same size, so there are only three different faces.½ mark
- Top and bottom (base): each is l × w, so together 2wl.½ mark
- Front and back: each is l × h, so together 2hl. The two side faces: each is w × h, so together 2hw.½ mark
- Add all six: TSA = 2wl + 2hl + 2hw = 2(wl + hl + hw).½ mark
TSA of a cuboid = 2(wl + hl + hw), where l, w, h are its length, width and height.
Check: Take a cuboid 3 cm × 2 cm × 1 cm. Faces: 6, 6, 3, 3, 2, 2, total 22 cm2. Formula: 2(6 + 3 + 2) = 22 cm2 ✓.
Answer to write in the exam
Top and bottom: 2 × l × w = 2wl
Front and back: 2 × l × h = 2hl
Two sides: 2 × w × h = 2hw
∴ TSA of a cuboid = 2(wl + hl + hw)
Common mistakes that cost marks
- Adding only three faces (lw + lh + wh) and forgetting that each face has an equal opposite face, so the total must be doubled.
- Mixing up surface area with volume and writing lwh. Area is in square units; volume is in cubic units.
- For an open box (no lid), still using 2(wl + hl + hw). An open box has only one l × w face.
How this can come in the exam
MCQ (1 mark)
The total surface area of a cuboid 5 cm × 4 cm × 3 cm is
- 60 cm2
- 94 cm2
- 47 cm2
- 120 cm2
Show answer
(B) 94 cm2
2(5 × 4 + 4 × 3 + 3 × 5) = 2(20 + 12 + 15) = 2 × 47 = 94 cm2.
Short answer (2 marks)
A box without a lid measures 30 cm × 20 cm × 10 cm. How much cardboard is needed to make it?
Show answer
No lid, so only one 30 × 20 face. Area = lw + 2(lh + wh) = 600 + 2(300 + 200) = 600 + 1000 = 1600 cm2.Try one yourself
Find the total surface area of a cuboid 8 m long, 6 m wide and 2 m high.
Show answer
2(48 + 16 + 12) = 2 × 76 = 152 m2.
More questions like this
- Compare the formula for the volume of a cuboid with the formula for the area of a rectangle (length × width).
- Try to work out for yourself why this model explains the formula for the volume of a cuboid, i.e.,
volume = area of base × height.
Write the formula for the surface area and the volume of a cube. - Observe how these two formulas are special cases of the formulas 2(wl + hl + hw) and whl, when w = h = l.
- Two solid objects are made from the same material: Cube A has side 6 cm and Cuboid B has dimensions 9 cm × 6 cm × 4 cm. Compare their volumes and surface areas and determine which object has a greater surface area. How is this useful in a real-life situation?
- The volume of a cube is 64 cm3. What is its total surface area?