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NCERT Solutions Ganita Prakash (Part 1) Chapter 1 –12Section 1.2 Cubic Numbers — In-text Questions

Book page 11 Updated on2026-09-05

Q1.
How many cubes of side 1 cm make a cube of side 2 cm?
Answer

8 cubes.

Each layer is 2 × 2 = 4 unit cubes
There are 2 such layers
Total = 2 × 2 × 2 = 23 = 8
Why we multiply three times: A cube has length, breadth and height all equal. Filling it with unit cubes means choosing 2 positions along each of the three directions, so the count is 2 × 2 × 2. This is exactly why n × n × n is called "n cubed".
Q2.
How many cubes of side 1 cm will make a cube of side 3 cm?
Answer

27 cubes.

Each layer is 3 × 3 = 9 unit cubes
There are 3 such layers
Total = 3 × 3 × 3 = 33 = 27
Tip: Doubling the side does not double the number of unit cubes — it multiplies it by 8, because each of the three directions doubles. Going from side 2 to side 3 raises the count from 8 to 27, not from 8 to 12.
Q3.
These numbers are called perfect cubes. Can you see why they are named so?
Answer

Because each of them counts the unit cubes that fill a cube.

1 = 1 × 1 × 1 → a cube of side 1
8 = 2 × 2 × 2 → a cube of side 2
27 = 3 × 3 × 3 → a cube of side 3
64 = 4 × 4 × 4 → a cube of side 4
Why the name carries over: The same thing happened with squares: 1, 4, 9, 16 are called squares because they count the unit squares in a square. Here the solid figure lends its name to the number. Sanskrit made the same choice — ghana means both the solid cube and the third power.
Q4.
Is 9 a cube?
Answer

No.

2 × 2 × 2 = 8
3 × 3 × 3 = 27
9 lies between 8 and 27, with no whole number in between

So 9 is not a perfect cube — and neither is any number from 10 to 26.

Why the gap is so wide: Cubes grow much faster than squares. Between 8 and 27 there is no room for another cube because there is no whole number between 2 and 3. That is why perfect cubes are far rarer than perfect squares: below 100 there are ten squares but only four cubes (1, 8, 27, 64).
Q5.
Can you estimate the number of unit cubes in a cube with an edge length of 4 units?
Answer

64 unit cubes.

Each square layer holds 4 × 4 = 16 unit cubes
There are 4 such layers
Total = 16 × 4 = 4 × 4 × 4 = 43 = 64
Tip: Thinking in layers is worth keeping. It turns a three-dimensional count into "area of one layer × number of layers", which is how volume is measured throughout your later work.
Q6.
Complete the table below. 1³ = 1, 11³ = 1331, 2³ = 8, 12³ = , 3³ = 27, 13³ = 2197, 4³ = 64, 14³ = 2744, 5³ = 125, 15³ = , 6³ = , 16³ = , 7³ = , 17³ = 4913, 8³ = , 18³ = 5832, 9³ = , 19³ = 6859, 10³ = , 20³ = . What patterns do you notice in the table above?
Answer

The completed table of cubes:

13 =1113 =1331
23 =8123 =1728
33 =27133 =2197
43 =64143 =2744
53 =125153 =3375
63 =216163 =4096
73 =343173 =4913
83 =512183 =5832
93 =729193 =6859
103 =1000203 =8000

Patterns worth naming:

  • Parity carries over. Odd numbers have odd cubes, even numbers have even cubes.
  • Every last digit occurs. Unlike squares, cubes end in all ten digits: 1, 8, 7, 4, 5, 6, 3, 2, 9, 0 for bases 1 to 10.
  • The units digit determines the base's units digit. A cube ending in 8 has a base ending in 2; ending in 7 → base ends in 3; ending in 3 → base ends in 7; ending in 2 → base ends in 8. The digits 0, 1, 4, 5, 6, 9 stay put.
  • Zeros triple. 103 = 1000, 203 = 8000 — one zero in the number becomes three in the cube.
  • Cubes are sparse. Only 20 cubes reach 8000, while there are 89 squares in the same range.
Why every digit appears: Cubing the digits 0 to 9 gives 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, whose last digits are 0, 1, 8, 7, 4, 5, 6, 3, 2, 9 — all ten, each exactly once. So a cube's last digit tells you the last digit of its cube root uniquely, which is not true for squares.
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