Cube Numbers — Definition, List, and Examples

What Is a Cube Number?

A cube number is the result of multiplying a whole number by itself twice more — three copies in total — written as $n \times n \times n$, or $n^3$. So $2^3 = 2 \times 2 \times 2 = 8$, and 8 is a cube number. The word "cube" is literal: $n^3$ counts the unit blocks inside a cube whose edge is $n$ units long.

The first cube numbers are:

$$1,\ 8,\ 27,\ 64,\ 125,\ 216,\ 343,\ 512,\ 729,\ 1000$$

Cube Numbers From 1 to 10

Every cube number comes from a base integer. Here is the full list from $1^3$ to $10^3$, the range most students are asked to know by heart.

Notice how fast they grow. The gap from $1$ to $8$ is 7; the gap from $9^3$ to $10^3$ is 271. Cubes pull away from each other far quicker than square numbers do, because each step multiplies by an extra factor of $n$.

Is a cube number the same as a perfect cube?

Yes. "Cube number" and "perfect cube" name the same thing: an integer that is some whole number multiplied by itself three times. You can explore this further in the article on the perfect cube.

How to Find a Cube Number

Finding a cube number takes one rule: multiply the base by itself, then by itself once more. There are two reliable ways to do it.

• Direct multiplication. Write the base three times and multiply left to right, one step at a time. For $7^3$: first $7 \times 7 = 49$, then $49 \times 7 = 343$.

• Square-then-multiply. Square the base, then multiply by the base once more. For $7^3$: $7^2 = 49$, then $49 \times 7 = 343$. This is often quicker, because most students already know the squares.

Either route gives the same result, and both make the third factor explicit — the step that stops "cubed" from being mistaken for "times three." To find a cube number the reverse way, starting from the result and recovering the base, you use the cube root.

Where Cube Numbers Come From

Cube numbers are one branch of the exponents family. Raising a number to the power 3 is called cubing it, and it sits one step beyond squaring.

• Squaring ($n^2$) measures the area of a square with side $n$.

• Cubing ($n^3$) measures the volume of a cube with edge $n$.

That second line is the whole idea. If you build a cube 4 units on every edge, the number of unit blocks packed inside is $4 \times 4 \times 4 = 64$ — so 64 is a cube number and it is also the volume of that cube. The concept links directly to the geometry of solid shapes, and to square numbers, which do the same job one dimension lower.

There is a second pattern hiding in the list. Add up consecutive odd numbers and cube numbers appear:

$$1 = 1$$ $$3 + 5 = 8$$ $$7 + 9 + 11 = 27$$ $$13 + 15 + 17 + 19 = 64$$

Each cube $n^3$ is a run of $n$ consecutive odd numbers. This is the kind of structure that makes cubes worth knowing, not just memorising.

Examples of Cube Numbers

The examples below move from a one-step calculation to a word problem. Work each one before reading the solution.

Example 1: Find the cube of 6

$$6^3 = 6 \times 6 \times 6$$ $$6 \times 6 = 36$$ $$36 \times 6 = 216$$

Final answer: $6^3 = 216$.

Example 2: Find the cube of 5

The quick instinct is to reach for $5 \times 3 = 15$, treating "cubed" as "times three."

$$5^3 \ne 15$$

That result is wrong, and you can see why at a glance: $5^2$ is already $25$, so $5^3$ must be larger than 25, not smaller. Cubing means three copies multiplied, not added or tripled.

$$5^3 = 5 \times 5 \times 5$$ $$5 \times 5 = 25$$ $$25 \times 5 = 125$$

Final answer: $5^3 = 125$.

Example 3: Is 100 a cube number?

Test the whole numbers near the answer.

$$4^3 = 64$$ $$5^3 = 125$$

100 sits between 64 and 125, and there is no integer between 4 and 5.

Final answer: 100 is not a cube number.

Example 4: Find the cube of the negative number $-3$

A negative base keeps its sign when cubed, because three negative factors multiply to a negative.

$$(-3)^3 = (-3) \times (-3) \times (-3)$$ $$(-3) \times (-3) = 9$$ $$9 \times (-3) = -27$$

Final answer: $(-3)^3 = -27$.

[IMAGE PROMPT: A number line running from -30 to 30 with two points marked: +27 labeled "3³" and -27 labeled "(-3)³", mirrored across zero. Caption: "Cubing a negative number gives a negative result — the sign carries through." Alt text: "Number line showing plus 27 and minus 27 placed symmetrically around zero."]

Example 5: Which cube number is closest to 500?

List the cubes around 500.

$$7^3 = 343$$ $$8^3 = 512$$

Compare the distances.

$$500 - 343 = 157$$ $$512 - 500 = 12$$

Final answer: 512 is the closest cube number to 500.

Example 6: A storage crate is a perfect cube. Its volume is 729 cubic centimetres. How long is each edge?

The volume of a cube is edge cubed, so you need the base number whose cube is 729.

$$\text{edge}^3 = 729$$ $$9^3 = 729$$

Finding the base from the cube is the job of the cube root — the reverse operation.

Final answer: each edge is 9 cm long.

Properties of Cube Numbers

Cube numbers follow a handful of patterns that make them easy to recognise and check. Knowing these turns cubing from memorised arithmetic into something you can reason about.

• Odd stays odd, even stays even. The cube of an even number is even ($4^3 = 64$); the cube of an odd number is odd ($3^3 = 27$). The parity of the base always survives.

• Negative bases give negative cubes. Three negative factors multiply to a negative, so $(-n)^3 = -n^3$ — for example $(-4)^3 = -64$. This is the key difference from square numbers, which are never negative.

• The unit digit is predictable. A cube's last digit is fixed by the base's last digit. Bases ending in $0, 1, 4, 5, 6, 9$ keep that digit; a base ending in $2$ gives a cube ending in $8$ (and $8 \to 2$), while $3 \to 7$ (and $7 \to 3$).

• Each cube is a run of consecutive odd numbers. As shown above, $n^3$ equals a block of $n$ consecutive odd numbers ($27 = 7 + 9 + 11$).

• The sum of the first $n$ cubes is a perfect square. Add the cubes in order and the total is the square of the corresponding triangular number:

$$1^3 + 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$

For $n = 3$: $1 + 8 + 27 = 36 = 6^2$. This links cube numbers straight back to perfect squares, a result worth carrying into later algebra.

Why "Consecutive Odd Numbers" Reveals the Real Structure

The reason cube numbers matter is that they are the volume of physical space, and volume is where unit errors become expensive. When engineers size a container, a reservoir, or a reaction vessel, they are working in cubes: double the edge of a tank and you do not double the capacity — you multiply it by eight, because $2^3 = 8$.

That eight-fold jump is exactly what trips people up in the real world:

• Concrete and material orders scale as the cube of the linear size. A wall twice as thick as planned needs eight times the volume of fill, not twice.

• Cooling and heating depend on volume-to-surface ratios, both governed by cubes and squares — which is why small animals lose heat faster than large ones.

• 3D printing and packing count unit cells, so the block-stacking picture is literal: a cube $n$ on a side always uses $n^3$ cells.

The odd-number pattern ($1$, then $3+5$, then $7+9+11$) is not a party trick. It is the reason cubes can be built up one shell at a time, and it is used in number theory) to prove results about sums of cubes without ever computing a single product.

Common Mistakes With Cube Numbers

Mistake 1: Treating "cubed" as "multiply by 3"

Where it slips in: the first time a student meets the exponent 3, before the block picture is firm.

Don't do this: writing $5^3 = 15$ or $7^3 = 21$.

The correct way: cubing is three copies multiplied — $5^3 = 5 \times 5 \times 5 = 125$. Students first meeting cubes almost always reach for tripling; the fix is to check against the square, since $n^3$ must be larger than $n^2$.

Mistake 2: Multiplying only twice

Where it slips in: rushing the arithmetic and stopping one step early.

Don't do this: computing $4^3$ as $4 \times 4 = 16$ and calling it done.

The correct way: carry the third factor. $4 \times 4 = 16$, then $16 \times 4 = 64$. The habit that fixes this is writing all three factors on the line before multiplying, so the eye sees three numbers, not two.

Mistake 3: Confusing a cube with a cube root

Where it slips in: questions phrased as "find the cube number that gives 64."

Don't do this: cubing 64 to get a huge number when the question wanted the base.

The correct way: read the direction. A cube goes $n \to n^3$; a cube root goes $n^3 \to n$.

Practice Questions

Try these before checking the answers. Rebuild any cube from the block picture if it looks wrong.

1. Find the cube of 8.

2. Is 216 a cube number? If so, which base gives it?

3. Evaluate $(-5)^3$.

4. Which cube number lies closest to 700?

5. Use the pattern to find the last digit of $12^3$ without full multiplication.

6. Find the value of $1^3 + 2^3 + 3^3 + 4^3$ and show it is a perfect square.

Answers

1. $8^3 = 512$.

2. Yes; $6^3 = 216$, so the base is 6.

3. $(-5)^3 = -125$.

4. $8^3 = 512$ and $9^3 = 729$; since $729 - 700 = 29$ and $700 - 512 = 188$, the closest is 729.

5. The base ends in 2, so the cube ends in 8 (the full value is 1728).

6. $1 + 8 + 27 + 64 = 100 = 10^2$, matching $\left(\frac{4 \times 5}{2}\right)^2 = 10^2$.

Conclusion

• A cube number is an integer multiplied by itself three times: $n^3 = n \times n \times n$.

• The first ten cubes are $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$.

• Each cube $n^3$ is the volume of a cube with edge $n$, and equals a run of $n$ consecutive odd numbers.

• Cubing a negative base gives a negative cube; the finding-the-base direction is the cube root.

• The most common error is treating "cubed" as tripling — always check against the square.

To take cube numbers further with a teacher, explore Bhanzu's algebra tutor, help with algebra, or math classes online. Practice these problems to solidify your understanding, and if a cube ever looks wrong, rebuild it from the block picture. Want to work through cubes and their roots with a live Bhanzu trainer? Book a free demo class.

Read More

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• Cube of a binomial expansion

• Sum of cubes of n natural numbers

• Exponent rules for powers

• Cubic polynomials explained