Observe how these two formulas are special cases of the formulas 2(wl + hl + hw) and whl, when w = h = l.
Answer: Put w = h = l = a: 2(a2 + a2 + a2) = 6a2 and a × a × a = a3. These are the cube’s surface area and volume.
Step-by-step solution
Idea: A cube is a cuboid whose three edges are equal, so replace l, w and h all by the side a.
- Let w = h = l = a (the side of the cube).½ mark
- Surface area: 2(wl + hl + hw) = 2(a × a + a × a + a × a) = 2 × 3a2 = 6a2.1 mark
- Volume: whl = a × a × a = a3.½ mark
With w = h = l = a: 2(wl + hl + hw) = 6a² and whl = a³, the surface area and volume of a cube.
Check: Side 5: cuboid formula 2(25 + 25 + 25) = 150 and 6 × 25 = 150 ✓; 5 × 5 × 5 = 125 = 53 ✓.
Answer to write in the exam
Put w = h = l = a
TSA = 2(a2 + a2 + a2) = 6a2
Volume = a × a × a = a3
∴ Cube: TSA = 6a2, volume = a3
Common mistakes that cost marks
- Writing 2 × 3a = 6a instead of 2 × 3a2 = 6a2: each face area is a × a = a2.
- Writing a × a × a = 3a.
How this can come in the exam
MCQ (1 mark)
The total surface area of a cube is 216 cm2. Its side is
- 36 cm
- 6 cm
- 9 cm
- 12 cm
Show answer
(B) 6 cm
6a2 = 216 ⇒ a2 = 36 ⇒ a = 6 cm.
Try one yourself
Find the volume of a cube whose total surface area is 486 cm2.
Show answer
6a2 = 486 ⇒ a2 = 81 ⇒ a = 9 cm; volume = 93 = 729 cm3.
More questions like this
- 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?
- 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.