Perfect Cube Calculator - Check Numbers and Find Cube Roots
Instantly determine if any integer is a perfect cube, find its integer cube root, or locate the nearest perfect cubes.
Enter any integer — positive, negative, or zero — to check if it is a perfect cube. The calculator also shows the nearest perfect cubes for non-cube inputs.
Perfect Cube Calculator - Check Numbers and Find Cube Roots
Instantly determine if any integer is a perfect cube, find its integer cube root, or locate the nearest perfect cubes.
About the perfect cube calculator
A perfect cube is an integer that equals another integer multiplied by itself three times. In mathematical notation, a number n is a perfect cube if there exists an integer x such that x³ = n. The integer x is called the cube root of n, written as ∛n. The sequence of positive perfect cubes starts at 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, and continues indefinitely as each consecutive integer is cubed.
One of the most important features of cubes — unlike squares — is that negative numbers can also be perfect cubes. Because a negative number multiplied by itself an odd number of times remains negative, (−x)³ = −(x³). So −8 is a perfect cube (since (−2)³ = −8), −27 is a perfect cube (since (−3)³ = −27), and so on. The cube root of a negative perfect cube is negative. This is in contrast to square roots, where the square root of a negative number is not a real number.
Zero is also a perfect cube: 0³ = 0, and the cube root of zero is zero. This is consistent with the mathematical definition and is handled correctly by this calculator.
To check whether a number is a perfect cube by hand, you can use prime factorisation. Find all the prime factors of the number. If every prime factor appears exactly three times (or a multiple of three times) in the factorisation, the number is a perfect cube. For example, 216 = 2³ × 3³, so both prime factors appear exactly three times, confirming 216 is a perfect cube with cube root 6. For 72 = 2³ × 3², the factor 3 appears only twice — not in a group of three — so 72 is not a perfect cube.
Perfect cubes appear in many practical and theoretical contexts. In geometry, if you know the volume of a cube and the volume is a perfect cube, you can immediately determine the side length by taking the cube root. In physics, cubic relationships arise in volume scaling — doubling the side length of a cube increases its volume by a factor of eight (2³). In computer science, cubic complexity O(n³) describes algorithms whose run time grows proportionally to the cube of the input size, which is important when analysing nested loops.
This calculator determines whether your input is a perfect cube using exact integer arithmetic. If it is, the calculator displays the integer cube root along with the verification equation. If it is not, the calculator finds the nearest perfect cubes below and above the input so you can see where your number sits in the sequence.
Perfect cube examples
Four examples showing perfect cubes, non-perfect cubes, and negative perfect cubes.
| Number | Is Perfect Cube? | Details |
|---|---|---|
| 27 | Yes | 3³ = 3 × 3 × 3 = 27. The integer cube root is 3. A cube with side length 3 cm has a volume of 27 cm³. |
| 30 | No | 30 lies between 27 (3³) and 64 (4³). Its cube root is approximately 3.107, which is not an integer, so 30 is not a perfect cube. |
| −64 | Yes | (−4)³ = (−4) × (−4) × (−4) = −64. The integer cube root is −4. Negative integers can be perfect cubes. |
| 1000 | Yes | 10³ = 10 × 10 × 10 = 1000. The integer cube root is 10. This is the smallest four-digit perfect cube. |
How to use the perfect cube calculator
- Type any integer — positive, negative, or zero — into the Number input field.
- Click Calculate. The result appears instantly, telling you whether the number is a perfect cube.
- If it is a perfect cube, the integer cube root is displayed along with the verification equation x³ = n.
- If it is not a perfect cube, the nearest perfect cubes below and above your number are shown.
- Click Reset to clear the field, or tap one of the example buttons to load a pre-filled test case.
Perfect cube calculator FAQ
What is a perfect cube?
A perfect cube is an integer n for which there exists another integer x such that x³ = n. Examples include 1 (1³), 8 (2³), 27 (3³), and −125 ((−5)³). The integer x is called the integer cube root of n.
Can a negative number be a perfect cube?
Yes. Unlike square roots, cube roots of negative numbers are real and negative. For example, (−3)³ = −27, so −27 is a perfect cube with cube root −3. The rule is (−x)³ = −(x³), so every positive perfect cube has a corresponding negative perfect cube.
How do I check if a number is a perfect cube by hand?
Use prime factorisation. Find the prime factors of the number's absolute value. If every prime factor appears in the factorisation a number of times that is a multiple of 3, the number is a perfect cube. For example, 216 = 2³ × 3³ — both factors appear 3 times — so 216 is a perfect cube (6³ = 216).
Is 0 a perfect cube?
Yes. 0³ = 0 × 0 × 0 = 0, so zero is a perfect cube. Its integer cube root is 0. This is consistent with the standard mathematical definition of a perfect cube.
What is the difference between a perfect cube and a perfect square?
A perfect square is a number that equals an integer squared (x² = n), while a perfect cube is a number that equals an integer cubed (x³ = n). Some numbers are both — 64 is 8² and 4³. The key practical difference is that perfect squares must be non-negative, whereas perfect cubes can be negative.
What are the first ten positive perfect cubes?
The first ten positive perfect cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. They correspond to 1³ through 10³. The gaps between consecutive cubes grow larger as n increases because the cubing function is a rapidly increasing polynomial.