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?
Step-by-step solution
To find: Volumes and surface areas of A and B; which has more surface area; a real-life use
Idea: Use a3 and 6a2 for the cube, and lwh and 2(lw + wh + lh) for the cuboid. Equal volumes can still have different surface areas.
- Cube A: volume = a3 = 63 = 216 cm3.½ mark
- Cube A: surface area = 6a2 = 6 × 36 = 216 cm2.½ mark
- Cuboid B: volume = l × w × h = 9 × 6 × 4 = 216 cm3.½ mark
- Cuboid B: surface area = 2(lw + wh + lh) = 2[(9 × 6) + (6 × 4) + (4 × 9)] = 2(54 + 24 + 36) = 2 × 114 = 228 cm2.1 mark
- Same volume (216 cm3), but 228 > 216, so Cuboid B has the greater surface area.½ mark
- Real life: for the same storage capacity, a cube needs less covering material (cheaper packaging), while a cuboid exposes more area to its surroundings (it heats up or cools down faster). This matters in packaging, heat transfer and engineering.1 mark
Check: Count of 1 cm cubes: 6 × 6 × 6 = 216 and 9 × 6 × 4 = 216, the same ✓. Faces of B: 54 + 54 + 24 + 24 + 36 + 36 = 228 ✓.
Answer to write in the exam
Cube A: V = 63 = 216 cm3; SA = 6 × 62 = 216 cm2
Cuboid B: V = 9 × 6 × 4 = 216 cm3
SA = 2[(9 × 6) + (6 × 4) + (4 × 9)] = 2 × 114 = 228 cm2
Volumes are equal; 228 cm2 > 216 cm2
∴ Cuboid B has the greater surface area.
Use: for the same capacity a cube needs less packing material; a cuboid exposes more area to its surroundings (faster heating/cooling).
Common mistakes that cost marks
- Thinking equal volumes must mean equal surface areas. Shape changes the surface area.
- Forgetting the 2 in 2(lw + wh + lh) and getting 114 cm2 for the cuboid.
- Writing surface area in cm3 or volume in cm2.
How this can come in the exam
Assertion (A): A cube of side 4 cm and a cuboid 8 cm × 4 cm × 2 cm hold the same volume.
Reason (R): Of all cuboids with the same volume, the cube has the least surface area.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(B) Both A and R are true, but R is not the correct explanation of A.
A: 64 cm3 and 8 × 4 × 2 = 64 cm3, true. R is true (cube 96 cm2, cuboid 2(32 + 8 + 16) = 112 cm2), but R is about surface area and does not explain why the volumes are equal.
A sweet shop packs 1000 cm3 of sweets either in a cube-shaped box of side 10 cm or in a cuboid box 20 cm × 10 cm × 5 cm.
(i) Show that both boxes hold the same amount. (ii) Find the cardboard needed for each box (closed). (iii) Which box should the shop choose to save cardboard, and by how much?
Show answer
(i) 103 = 1000 cm3 and 20 × 10 × 5 = 1000 cm3 (1 mark). (ii) Cube: 6 × 100 = 600 cm2; cuboid: 2(200 + 50 + 100) = 700 cm2 (2 marks). (iii) The cube saves 700 − 600 = 100 cm2 per box (1 mark).Try one yourself
Compare the surface areas of a cube of side 12 cm and a cuboid 18 cm × 12 cm × 8 cm. Do they have the same volume?
Show answer
Volumes: 1728 cm3 and 1728 cm3, equal. Surface areas: 6 × 144 = 864 cm2 and 2(216 + 96 + 144) = 912 cm2. The cuboid has more.
More questions like this
- The volume of a cube is 64 cm3. What is its total surface area?
- How many small cubes with side 20 cm can be packed tight in a cubical box with side 2 m?
- The dimensions of a godown are 40 m × 25 m × 10 m. If it is filled with cuboidal boxes, each of dimensions 2 m × 1.25 m × 1 m, then find the number of boxes.
- Two cubes each of volume 125 cm3 are joined end to end. Find the surface area of the resulting cuboid.
- A cube of side 4 cm is cut into cubes of side 1 cm. What is the ratio of the surface areas of the original cube and all the cut-out cubes? (Note that there is no change in volume but a big change in the surface area. This property has major consequences in the biological world.)