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.
- 1. 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.
- 2. Write the formula for the surface area and the volume of a cube.
Step-by-step solution
Idea: Think of a cuboid as a pack of cards: every card is the same rectangle (the base), and the pack is as tall as the cuboid.
1. 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.
- One layer of unit cubes on the base has l × w cubes, because the base is an l × w rectangle. That is the area of the base.½ mark
- Stacking layers like cards, each layer is exactly the same rectangle. A height of h units holds h such layers.½ mark
- Total cubes = (cubes in one layer) × (number of layers) = (l × w) × h. So volume = area of base × height = lwh.1 mark
2. Write the formula for the surface area and the volume of a cube.
- A cube is a cuboid with l = w = h = a. Surface area = 2(a × a + a × a + a × a) = 2 × 3a2 = 6a2.½ mark
- Volume = a × a × a = a3.½ mark
Check: A 4 × 3 × 2 box: one layer has 4 × 3 = 12 unit cubes, 2 layers give 24, and 4 × 3 × 2 = 24 ✓. A cube of side 2: 6 faces of 4 = 24 = 6 × 22 ✓, and 8 unit cubes = 23 ✓.
Answer to write in the exam
1.
Cubes in one layer = l × w = area of base
Number of layers = h
Volume = (l × w) × h
∴ Volume of cuboid = area of base × height = lwh
2.
Total surface area of cube = 6a2
Volume of cube = a3
Common mistakes that cost marks
- Thinking the base must be the bottom face only. Any face can be the base, as long as the height is measured at right angles to it.
- Writing the surface area of a cube as 4a2 (only the four side faces) instead of 6a2.
- Writing the volume of a cube as 3a instead of a3.
How this can come in the exam
The base of a cuboid has area 35 cm2 and its height is 6 cm. Its volume is
- 41 cm3
- 210 cm3
- 105 cm3
- 70 cm3
Show answer
(B) 210 cm3
Volume = area of base × height = 35 × 6 = 210 cm3.
Assertion (A): A cube of side 3 cm has volume 27 cm3.
Reason (R): The surface area of a cube of side a is 6a2.
- 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: 33 = 27, true. R is true too, but it is about surface area, so it does not explain the volume.
Try one yourself
A cube has side 7 cm. Find its total surface area and its volume.
Show answer
TSA = 6 × 49 = 294 cm2; volume = 73 = 343 cm3.
More questions like this
- 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?
- 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.