Ganita Manjari · Class 9 · Part II · Chapter 14
All 8 questions on cuboids and cubes, including the painted cube and integer cuboids of surface area 100 cm². Every answer is checked a second way.
Brilliant! You have finished every question in this exercise.
Total surface area: the sum of the areas of all faces.
Volume: the number of unit cubes that fit inside.
Cube: a cuboid with equal length, width and height.
The total surface area is 96 cm².
2 m = 200 cm. Along each edge of the box the number of small cubes is 20020 = 10.
Check with volumes: 200³20³ = 8,000,0008,000 = 1000 ✓.
1000 small cubes fit.
The boxes fit exactly: 402 = 20 along the length, 251.25 = 20 along the width and 101 = 10 along the height, and 20 × 20 × 10 = 4000 ✓.
The godown holds 4000 boxes.
Joined end to end, the cuboid is 10 cm long, 5 cm wide and 5 cm high.
Check: the two cubes have 12 faces of 25 cm² = 300 cm². The two faces that were joined are hidden: 300 − 2 × 25 = 250 ✓.
The surface area is 250 cm².
| Solid | Number | Surface area of each | Total |
|---|---|---|---|
| Original cube (4 cm) | 1 | 6 × 4² = 96 cm² | 96 cm² |
| Small cubes (1 cm) | 4³ = 64 | 6 × 1² = 6 cm² | 64 × 6 = 384 cm² |
The volume did not change, but the surface area became 4 times larger. Cutting into smaller pieces always exposes more surface. This is why a fine powder dissolves faster than a lump, and why small cells can absorb food quickly.
The ratio is 1 : 4.
Let the edges be l, w, h. The three face areas are lw = 6, wh = 15 and lh = 10.
The edges are l = 2, w = 3, h = 5: lw = 6 ✓, wh = 15 ✓, lh = 10 ✓, and 2 × 3 × 5 = 30 ✓.
The volume is 30 cm³.
| Type of small cube | Where it is | How many | Painted faces |
|---|---|---|---|
| Corner | 8 corners | 8 | 3 |
| Edge (not a corner) | 12 edges, each has 5 − 2 = 3 of them | 12 × 3 = 36 | 2 |
| Face (not an edge) | 6 faces, each has 3 × 3 = 9 | 6 × 9 = 54 | 1 |
| Inside | a 3 × 3 × 3 block | 27 | 0 |
(i) 8, (ii) 36, (iii) 54, (iv) 27.
2(lw + lh + wh) = 100, so lw + lh + wh = 50. Take l ≤ w ≤ h.
For the smallest edge l, add l² to both sides and factorise:
Since l is the smallest, lw + lh + wh ≥ 3l², so 3l² ≤ 50, so l ≤ 4. Try l = 1, 2, 3, 4.
| l | (w + l)(h + l) = 50 + l² | Factor pairs with w + l ≥ 2l | Cuboid |
|---|---|---|---|
| 1 | 51 = 3 × 17 | w + 1 = 3, h + 1 = 17 | w = 2, h = 16: 1 × 2 × 16 |
| 2 | 54 = 6 × 9 | w + 2 = 6, h + 2 = 9 | w = 4, h = 7: 2 × 4 × 7 |
| 3 | 59 (prime) | none | none |
| 4 | 66 = 6 × 11, 2 × 33, 3 × 22 | need w + 4 ≥ 8: none | none |
(i) Yes, there are two. (ii) and (iii) The table covers every l from 1 to 4, and l cannot be larger, so these are all.
Check: 1 × 2 × 16: 2(2 + 16 + 32) = 100 ✓. 2 × 4 × 7: 2(8 + 14 + 28) = 100 ✓.
There are exactly two such cuboids: 1 × 2 × 16 and 2 × 4 × 7.
Units must match before any formula is used.
| Question | What is asked | Key idea | Answer |
|---|---|---|---|
| Q1 | Cube of volume 64 | a³ = 64 | 96 cm² |
| Q2 | Small cubes in a box | 200 over 20 cubed | 1000 |
| Q3 | Boxes in a godown | 10,000 over 2.5 | 4000 |
| Q4 | Two joined cubes | 10 × 5 × 5 | 250 cm² |
| Q5 | Cut a cube | 96 and 384 | 1 : 4 |
| Q6 | Volume from face areas | (lwh)² = 900 | 30 cm³ |
| Q7 | Painted cube | Corners, edges, faces, inside | 8, 36, 54, 27 |
| Q8 | Surface area 100 cm² | (w + l)(h + l) = 50 + l² | 1×2×16 and 2×4×7 |
Source of the questions: NCERT, Ganita Manjari, Grade 9, Part II, Chapter 14, Exercise Set 14.1. The solutions, explanations and diagrams on this page are our own working.