Perfect cubes are the numbers you get by cubing whole numbers: 1, 8, 27, 64, 125 and so on. They come up when simplifying cube roots, factoring sums and differences of cubes, and working with volumes. This generator lists the cubes of any range of whole numbers, negative ones included, or every perfect cube up to a value you choose, along with the gap from each cube to the next.
How to use the perfect cubes list
- Choose n³ for n from … to … to list a range, or Every perfect cube up to a value to list all cubes from 0 to a limit.
- Enter the range (for example −5 to 20) or the largest value (for example 10,000).
- Read the table of n, n³ and the difference from the previous cube. The tape shows the count, the first and last cube, and their exact sum. Up to 2,000 rows can be listed at once, for n up to one billion in size.
Cube formulas
The gaps 1, 7, 19, 37, 61, … are the centered hexagonal numbers. The sum of the first n cubes has a remarkably neat closed form:
Perfect cubes from 1 to 20
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 11 | 1,331 |
| 2 | 8 | 12 | 1,728 |
| 3 | 27 | 13 | 2,197 |
| 4 | 64 | 14 | 2,744 |
| 5 | 125 | 15 | 3,375 |
| 6 | 216 | 16 | 4,096 |
| 7 | 343 | 17 | 4,913 |
| 8 | 512 | 18 | 5,832 |
| 9 | 729 | 19 | 6,859 |
| 10 | 1,000 | 20 | 8,000 |
Worked example: cubes up to 1,000
Choosing “every perfect cube up to 1,000” gives ⌊∛1,000⌋ = 10, so the list is 0, 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1,000: eleven cubes. Their sum is (10 × 11 ÷ 2)² = 55² = 3,025, matching the total on the tape.
A range that crosses zero shows the symmetry of cubes: listing n = −3 to 3 gives −27, −8, −1, 0, 1, 8, 27, and the sum is 0 because each negative cube cancels its positive partner.
Using the list
Simplifying cube roots
To simplify ∛54, look for the largest perfect cube that divides 54. Since 27 × 2 = 54, ∛54 = ∛27 × ∛2 = 3∛2. The simplify radicals calculator does this for any index.
Factoring sums and differences of cubes
Recognizing cubes lets you use the identities a³ − b³ = (a − b)(a² + ab + b²) and a³ + b³ = (a + b)(a² − ab + b²). For example, x³ − 64 = (x − 4)(x² + 4x + 16) because 64 = 4³.
Estimating cube roots
∛500 lies between 7 and 8 because 343 < 500 < 512, and since 500 is very close to 512, the root is just under 8 (about 7.937).
Cubes and volume
A cube-shaped box with whole-number edges holds a perfect-cube number of unit cubes. That is why many standard volumes are cubes: 1 ft³ = 12³ = 1,728 in³ and 1 m³ = 100³ = 1,000,000 cm³.
For a single value use the cube number calculator, and to reverse it use the cube root calculator.
Frequently asked questions
What is a perfect cube?
A perfect cube is a whole number raised to the third power, such as 1, 8, 27, 64 and 125. Negative whole numbers have perfect cubes too: (−2)³ = −8.
How many perfect cubes are there up to 1,000?
Eleven if you include 0: 0, 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1,000. The count up to N is ⌊∛N⌋ + 1.
How can I tell if a number is a perfect cube?
Factor it into primes. It is a perfect cube when every prime appears a multiple of three times, as in 5,832 = 2³ × 3⁶ = 18³. The cube root calculator checks this instantly.
What is the sum of the first n cubes?
1³ + 2³ + … + n³ = [n(n + 1)/2]². The sum of the first n cubes is always the square of the n-th triangular number, so 1 + 8 + 27 + 64 = 100 = 10².