What is a perfect cube?
A perfect cube is a number you get by multiplying an integer by itself three times. 8 is a perfect cube because 2 × 2 × 2 = 8, and we write that as 2³. The number being multiplied is called the cube root.
n is a perfect cube if n = k × k × k for some whole number k The word "cube" comes from geometry: a cube with sides of 4 units has a volume of 4 × 4 × 4 = 64 cubic units, so 64 is a perfect cube. Any whole-number side length gives a perfect cube volume, and any perfect cube volume can be built as a cube with whole-number sides.
Numbers that aren't perfect cubes still have cube roots — they're just not integers. The cube root of 100 is about 4.641589, which sits between the cube roots of 64 and 125.
Is 1 a perfect cube?
Yes. 1 × 1 × 1 = 1, so 1 = 1³ and 1 is a perfect cube. It's the smallest positive one.
It's an easy one to doubt, because multiplying by 1 doesn't change anything — so it can feel like nothing happened. But the definition only asks whether some whole number cubed gives the result, and 1 does. For the same reason 1 is also a perfect square, a perfect fourth power, and a perfect power of every kind.
Zero is a perfect cube too: 0 × 0 × 0 = 0, so 0 = 0³. And −1 is a perfect cube, since (−1)³ = −1.
List of perfect cubes
The perfect cubes of 1 to 30:
| n | Notation | Perfect cube |
|---|---|---|
| 1 | 1³ | 1 |
| 2 | 2³ | 8 |
| 3 | 3³ | 27 |
| 4 | 4³ | 64 |
| 5 | 5³ | 125 |
| 6 | 6³ | 216 |
| 7 | 7³ | 343 |
| 8 | 8³ | 512 |
| 9 | 9³ | 729 |
| 10 | 10³ | 1,000 |
| 11 | 11³ | 1,331 |
| 12 | 12³ | 1,728 |
| 13 | 13³ | 2,197 |
| 14 | 14³ | 2,744 |
| 15 | 15³ | 3,375 |
| 16 | 16³ | 4,096 |
| 17 | 17³ | 4,913 |
| 18 | 18³ | 5,832 |
| 19 | 19³ | 6,859 |
| 20 | 20³ | 8,000 |
| 21 | 21³ | 9,261 |
| 22 | 22³ | 10,648 |
| 23 | 23³ | 12,167 |
| 24 | 24³ | 13,824 |
| 25 | 25³ | 15,625 |
| 26 | 26³ | 17,576 |
| 27 | 27³ | 19,683 |
| 28 | 28³ | 21,952 |
| 29 | 29³ | 24,389 |
| 30 | 30³ | 27,000 |
Some larger round ones:
| n | Notation | Perfect cube |
|---|---|---|
| 40 | 40³ | 64,000 |
| 50 | 50³ | 125,000 |
| 60 | 60³ | 216,000 |
| 70 | 70³ | 343,000 |
| 80 | 80³ | 512,000 |
| 90 | 90³ | 729,000 |
| 100 | 100³ | 1,000,000 |
Negative perfect cubes mirror the positive ones exactly, which is the main way cubes differ from squares:
| n | Notation | Perfect cube |
|---|---|---|
| −1 | (−1)³ | −1 |
| −2 | (−2)³ | −8 |
| −3 | (−3)³ | −27 |
| −4 | (−4)³ | −64 |
| −5 | (−5)³ | −125 |
| −6 | (−6)³ | −216 |
| −7 | (−7)³ | −343 |
| −8 | (−8)³ | −512 |
| −9 | (−9)³ | −729 |
| −10 | (−10)³ | −1,000 |
How to tell if a number is a perfect cube
Short of cubing numbers until you overshoot, there are three practical methods.
1. Prime factorisation
This is the reliable one. Break the number into prime factors with a Factor Calculator; it's a perfect cube exactly when every prime appears a number of times divisible by 3.
- 1728: 2⁶ × 3³. The exponents are 6 and 3, both multiples of 3, so it's a perfect cube. Divide each exponent by 3 to get the root: 2² × 3 = 12, and indeed 12³ = 1728.
- 200: 2³ × 5². The 5 appears twice, which isn't a multiple of 3, so 200 is not a perfect cube.
2. The last-digit check
For integer cubes, each possible last digit of the cube corresponds to exactly one possible last digit of its integer cube root.
| Cube ends in | Root ends in |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
| 3 | 7 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 3 |
| 8 | 2 |
| 9 | 9 |
Note what this rule can and can't do. Unlike squares — where a number ending in 2, 3, 7 or 8 is definitely not a square — every digit is a possible cube ending, so this check can never rule a number out. What it does is tell you which root to try. Given 50653, the last digit 3 means the root ends in 7; since 30³ = 27000 and 40³ = 64000, the root is between 30 and 40, so the only candidate is 37 — and 37³ = 50653.
3. Estimate and check
Cubes grow fast, so there are few candidates in any range. For 9261: 20³ = 8000 and 21³ = 9261. Two guesses. This is what the calculator above does, using exact whole-number arithmetic so it stays correct for numbers far too large to check by hand.
Examples
- 1 = 1³
- 8 = 2³
- 27 = 3³
- 64 = 4³
- 125 = 5³
- −8 = (−2)³
- −27 = (−3)³
- 1728 = 12³
And some near misses: 100 is not a perfect cube (it falls between 4³ = 64 and 5³ = 125), nor is 1000000001, which sits just above 1000³ = 1000000000.
Perfect cubes and perfect squares
A number can be both. 64 is 8² and 4³; 729 is 27² and 9³. You can use the Square Root & Exponent Calculator to work with powers such as these. Numbers that are both are exactly the perfect sixth powers — 1, 64, 729, 4096, 15625 — since a number divisible by both 2 and 3 in all its exponents is divisible by 6.
The important difference is sign. Squaring always gives a positive result, so no negative number is a perfect square. Cubing preserves sign, so negative perfect cubes exist and every real number has exactly one real cube root.
Frequently asked questions
Can a negative number be a perfect cube?
Yes. −8 is a perfect cube because (−2) × (−2) × (−2) = −8. Multiplying three negatives together leaves a negative result, so cubing preserves the sign of the original number. This is the main way cubes differ from squares, where no negative number qualifies.
Is 1 a perfect cube?
Yes. 1 × 1 × 1 = 1, so 1 = 1³. It's the smallest positive perfect cube, and it's also a perfect square and a perfect power of every other kind.
Is 0 a perfect cube?
Yes. 0 × 0 × 0 = 0, so 0 = 0³. Zero is also its own cube root.
How many perfect cubes are there up to 1000?
Ten, counting 1000 itself: 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000. Perfect cubes thin out quickly — there are only 10 up to 1000, but 100 up to a million, because cubes grow as fast as their roots multiply.
Can a number be both a perfect square and a perfect cube?
Yes, and those numbers are the perfect sixth powers: 1, 64, 729, 4096, 15625 and so on. 64 is 8 squared and 4 cubed; 729 is 27 squared and 9 cubed.
Can a fraction or decimal be a perfect cube?
The term perfect cube is usually used for integers that are the cube of another integer. This calculator checks integers only. However, fractions and decimals can also be exact cubes: 8/27 is (2/3)³, and 0.008 is 0.2³.
What's the cube root of a number that isn't a perfect cube?
It's an irrational number — a decimal that never ends or repeats. The cube root of 2 is about 1.259921, and no fraction gives it exactly. The calculator shows an approximation along with the two integer cubes the value sits between.