Cubic Numbers — First Steps | Mathematics

Chapter 1 · A Square and A Cube

1.2 Cubic Numbers — First Steps

A square grows in two directions. A cube grows in three — and that one extra direction changes more than you'd expect, including what happens to a negative number.

🎯 By the end of this you can…

✓Say what makes a solid a cube
✓Work out how many unit cubes build a bigger cube, by counting layers
✓Recognise the pattern of cubic numbers: 1, 8, 27, 64, 125…
✓Explain why a negative number cubed stays negative — unlike squaring

💡 The idea

A cube is a solid where every side is equal, meeting at right angles

Just like a square is built from unit squares in a flat grid, a cube is built from unit CUBES stacked in three directions: along, across, and up.

🧊 How many 1 cm cubes fill a 2 cm cube?

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Each click adds one more 1 cm cube to the stack.

📋 Counting them without clicking one at a time

1
A cube of side $n$ has $n$ layers stacked up.
2
Each layer is an $n \times n$ grid of unit cubes.
3
Total unit cubes $= n \times n \times n = n^{3}$ .

🧊 How many 1 cm cubes fill a 3 cm cube?

0 / 27

Each click adds one more 1 cm cube to the stack.

✏️ Worked example

How many 1 cm cubes make a cube of side 3 cm?

1
Each of the 3 layers is a $3 \times 3 = 9$ grid of unit cubes.
2
Three layers: $9 \times 3 = 27$ unit cubes in total.

✅ Answer: $3^{3} = 27$

💡 The idea

The cubic numbers, so far

$1^{3} = 1, 2^{3} = 8, 3^{3} = 27, 4^{3} = 64, 5^{3} = 125$ — each one grows a lot faster than the matching square, because it's being multiplied by itself THREE times, not two.

⚠️ The classic trap

Squaring a negative number always gives a positive answer, because a negative times a negative is positive. CUBING does not work the same way — a negative number cubed stays negative, because there are three negative signs multiplied together, and an odd number of negatives multiplied stays negative.

🤔 What is (−2)³?

🎮 Your turn with cubic numbers

⭐ 0 XP🔥 streak 0

What is

6^{3}

What is

7^{3}

What number, cubed, gives 1000?

What is

(- 3)^{3}

How many 1 cm cubes would build a cube of side 4 cm?

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🏅

You finished this lesson!

You've met cubic numbers and seen why a cube's negative sign behaves differently from a square's. Finding a cube ROOT — reversing this, the way prime-factor pairing reversed squaring — is next, once more of this chapter is in front of you.

Keep reading — this lesson marks itself done automatically.